Digital Image Processing - profs.info.uaic.roancai/DIP/curs/DIP w1 2017.pdf · •R.C. Gonzales, R.E. Woods, S.L. Eddins, Digital Image Processing Using MATLAB, ... DIP –used in
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Digital Image Processing
Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Evaluation
1(base) + 3 (course test) + 3 (lab activity)+ 3 (article presentation)
gt 45
Weeks 1 ndash 7 ndash coursesWeek 8 ndash (course) testWeeks 9 ndash 12 ndash labWeeks 13 ndash 14 ndash article lab evaluationWeek 15 or 16 ndash article presentation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Bibliography
bull RC Gonzales RE Woods Digital Image Processing Prentice Hall2008 3rd ed
bull RC Gonzales RE Woods SL Eddins Digital Image Processing Using MATLAB Prentice Hall 2003
bull httpwwwimageprocessingplacecom
bullM Petrou C Petrou Image Processing the fundamentals John Wiley 2010 2nd ed
bull W Burger MJ Burge Digital Image Processing An Algorithmic Introduction Using Java Springer 2008
Digital Image ProcessingDigital Image Processing
Week 1Week 1
bull Image Processing Toolbox (httpwwwmathworkscomproductsimage)
bull C Solomon T Breckon Fundamentals of Digital Image Processing A Practical Approach with Examples in Matlab Wiley-Blackwell 2011
bullWK Pratt Digital Image Processing Wiley-Interscience 2007
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Meet LenaThe First Lady of the Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lenna Soderberg (Sjoumloumlblom) and Jeff Seideman taken in May 1997Imaging Science amp Technology Conference
Digital Image ProcessingDigital Image Processing
Week 1Week 1
What is Digital Image Processing
f(xy) = intensity gray level of the image at spatial point (xy)
x y f(xy) ndash finite discrete quantities -gt digital image
Digital Image Processing = processing digital images by means of a digital computer
A digital image is composed of a finite number of elements (location value of intensity)
These elements are called picture elements image elements pels pixels
( )i j ijx y f
3 f D
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image processing is not limited to the visual band of the electromagnetic (EM) spectrum
Image processing gamma to radio waves ultrasound electron microscopy computer-generated images
image processing image analysis computer vision
Image processing = discipline in which both the input and the output of a process are images
Computer Vision = use computer to emulate human vision (AI) ndashlearning making inferences and take actions based on visual inputs
Image analysis (image understanding) = segmentation partitioning images into regions or objects
(link between image processing and image analysis)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Distinction between image processing image analysis computer vision
low-level mid-level high-level processes
Low-level processes image preprocessing to reduce noise contrast enhancement image sharpening both input and output are images
Mid-level processes segmentation partitioning images into regions or objects description of the objects for computer processing classificationrecognition of individual objects inputs are generally images outputs are attributes extracted from the input image (eg edges contours identity of individual objects)
High-level processes ldquomaking senserdquo of a set of recognized objects performing the cognitive functions associated with vision
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing (Gonzalez + Woods) =
processes whose inputs and outputs are images +
processes that extract attributes from images recognition of individual objects
(low- and mid-level processes)
Example
automated analysis of text = acquiring an image containing text preprocessing the image (enhancement sharpening) extracting (segmenting) the individual characaters describing the characters in a form suitable for computer processing recognition of individual characters
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The Origins of DIP
Newspaper industry pictures were sent by submarine cable between London and New York
Before Bartlane cable picture transmission system (early 1920s) ndash 1 week
With Bartlane system less than 3 hours
Specialized printing equipment coded pictures for cable transmission and reconstructed them at the receiving end (1920s -5 distict levels of gray 1929 ndash 15 levels)
This example is not DIP the computer is not involved
DIP is linked and devolps in the same rhythm as digital computers (data storage display and transmission)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
A digital pictureproduced in 1921from a coded tapeby a telegraph printer withspecial type faces(McFarlane)
A digital picturemade in 1922from a tapepunched after the signals hadcrossed the Atlantic twice(McFarlane)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1964 Jet Propulsion Laboratory (Pasadena California) processed pictures of the moon transmitted by Ranger 7 (corrected image distortions)
The first picture of the moon by a US spacecraft Ranger 7 took this image July 31 1964 about 17 minutes before impacting the lunar surface(Courtesy of NASA)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1960-1970 ndash image processing techniques were used in medical image remote Earth resources observations astronomy
1970s ndash invention of CAT (computerized axial tomography)
CAT is a process in which a ring of detectors encircles an object (patient) and a X-ray source concentric with the detector ring rotates about the object The X-ray passes through the patient and the resulted waves are collected at the opposite end by the detectors As the source rotates the procedure is repeated Tomography consists of algorithms that use the sense data to construct an image that represent a ldquoslicerdquo through the object Motion of the object in a direction perpendicular to the ring of detectors produces a set of ldquoslicesrdquo which can be assembled in a 3D information of the inside of the object
Digital Image ProcessingDigital Image Processing
Week 1Week 1
loz geographers use DIP to study pollution patterns from aerial and satellite imagery
loz archeology ndash DIP allowed restoring blurred pictures that recorded rare artifacts lost or damaged after being photographed
loz physics ndash enhance images of experiments (high-energy plasmas electron microscopy)
loz astronomy biology nuclear medicine law enforcement industry
DIP ndash used in solving problems dealing with machine perception ndashextracting from an image information suitable for computer processing (statistical moments Fourier transform coefficients hellip)
loz automatic character recognition industrial machine vision for product assembly and inspection military recognizance automatic processing of fingerprints machine processing of aerial and satellite imagery for weather prediction Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of Fields that Use DIP
Images can be classified according to their sources (visual X-ray hellip)
Energy sources for images electromagnetic energy spectrum acoustic ultrasonic electronic computer- generated
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Electromagnetic waves can be thought as propagating sinusoidal
waves of different wavelength or as a stream of massless particles
each moving in a wavelike pattern with the speed of light Each
massless particle contains a certain amount (bundle) of energy Each
bundle of energy is called a photon If spectral bands are grouped
according to energy per photon we obtain the spectrum shown in the
image above ranging from gamma-rays (highest energy) to radio
waves (lowest energy)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Evaluation
1(base) + 3 (course test) + 3 (lab activity)+ 3 (article presentation)
gt 45
Weeks 1 ndash 7 ndash coursesWeek 8 ndash (course) testWeeks 9 ndash 12 ndash labWeeks 13 ndash 14 ndash article lab evaluationWeek 15 or 16 ndash article presentation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Bibliography
bull RC Gonzales RE Woods Digital Image Processing Prentice Hall2008 3rd ed
bull RC Gonzales RE Woods SL Eddins Digital Image Processing Using MATLAB Prentice Hall 2003
bull httpwwwimageprocessingplacecom
bullM Petrou C Petrou Image Processing the fundamentals John Wiley 2010 2nd ed
bull W Burger MJ Burge Digital Image Processing An Algorithmic Introduction Using Java Springer 2008
Digital Image ProcessingDigital Image Processing
Week 1Week 1
bull Image Processing Toolbox (httpwwwmathworkscomproductsimage)
bull C Solomon T Breckon Fundamentals of Digital Image Processing A Practical Approach with Examples in Matlab Wiley-Blackwell 2011
bullWK Pratt Digital Image Processing Wiley-Interscience 2007
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Meet LenaThe First Lady of the Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lenna Soderberg (Sjoumloumlblom) and Jeff Seideman taken in May 1997Imaging Science amp Technology Conference
Digital Image ProcessingDigital Image Processing
Week 1Week 1
What is Digital Image Processing
f(xy) = intensity gray level of the image at spatial point (xy)
x y f(xy) ndash finite discrete quantities -gt digital image
Digital Image Processing = processing digital images by means of a digital computer
A digital image is composed of a finite number of elements (location value of intensity)
These elements are called picture elements image elements pels pixels
( )i j ijx y f
3 f D
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image processing is not limited to the visual band of the electromagnetic (EM) spectrum
Image processing gamma to radio waves ultrasound electron microscopy computer-generated images
image processing image analysis computer vision
Image processing = discipline in which both the input and the output of a process are images
Computer Vision = use computer to emulate human vision (AI) ndashlearning making inferences and take actions based on visual inputs
Image analysis (image understanding) = segmentation partitioning images into regions or objects
(link between image processing and image analysis)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Distinction between image processing image analysis computer vision
low-level mid-level high-level processes
Low-level processes image preprocessing to reduce noise contrast enhancement image sharpening both input and output are images
Mid-level processes segmentation partitioning images into regions or objects description of the objects for computer processing classificationrecognition of individual objects inputs are generally images outputs are attributes extracted from the input image (eg edges contours identity of individual objects)
High-level processes ldquomaking senserdquo of a set of recognized objects performing the cognitive functions associated with vision
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing (Gonzalez + Woods) =
processes whose inputs and outputs are images +
processes that extract attributes from images recognition of individual objects
(low- and mid-level processes)
Example
automated analysis of text = acquiring an image containing text preprocessing the image (enhancement sharpening) extracting (segmenting) the individual characaters describing the characters in a form suitable for computer processing recognition of individual characters
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The Origins of DIP
Newspaper industry pictures were sent by submarine cable between London and New York
Before Bartlane cable picture transmission system (early 1920s) ndash 1 week
With Bartlane system less than 3 hours
Specialized printing equipment coded pictures for cable transmission and reconstructed them at the receiving end (1920s -5 distict levels of gray 1929 ndash 15 levels)
This example is not DIP the computer is not involved
DIP is linked and devolps in the same rhythm as digital computers (data storage display and transmission)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
A digital pictureproduced in 1921from a coded tapeby a telegraph printer withspecial type faces(McFarlane)
A digital picturemade in 1922from a tapepunched after the signals hadcrossed the Atlantic twice(McFarlane)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1964 Jet Propulsion Laboratory (Pasadena California) processed pictures of the moon transmitted by Ranger 7 (corrected image distortions)
The first picture of the moon by a US spacecraft Ranger 7 took this image July 31 1964 about 17 minutes before impacting the lunar surface(Courtesy of NASA)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1960-1970 ndash image processing techniques were used in medical image remote Earth resources observations astronomy
1970s ndash invention of CAT (computerized axial tomography)
CAT is a process in which a ring of detectors encircles an object (patient) and a X-ray source concentric with the detector ring rotates about the object The X-ray passes through the patient and the resulted waves are collected at the opposite end by the detectors As the source rotates the procedure is repeated Tomography consists of algorithms that use the sense data to construct an image that represent a ldquoslicerdquo through the object Motion of the object in a direction perpendicular to the ring of detectors produces a set of ldquoslicesrdquo which can be assembled in a 3D information of the inside of the object
Digital Image ProcessingDigital Image Processing
Week 1Week 1
loz geographers use DIP to study pollution patterns from aerial and satellite imagery
loz archeology ndash DIP allowed restoring blurred pictures that recorded rare artifacts lost or damaged after being photographed
loz physics ndash enhance images of experiments (high-energy plasmas electron microscopy)
loz astronomy biology nuclear medicine law enforcement industry
DIP ndash used in solving problems dealing with machine perception ndashextracting from an image information suitable for computer processing (statistical moments Fourier transform coefficients hellip)
loz automatic character recognition industrial machine vision for product assembly and inspection military recognizance automatic processing of fingerprints machine processing of aerial and satellite imagery for weather prediction Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of Fields that Use DIP
Images can be classified according to their sources (visual X-ray hellip)
Energy sources for images electromagnetic energy spectrum acoustic ultrasonic electronic computer- generated
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Electromagnetic waves can be thought as propagating sinusoidal
waves of different wavelength or as a stream of massless particles
each moving in a wavelike pattern with the speed of light Each
massless particle contains a certain amount (bundle) of energy Each
bundle of energy is called a photon If spectral bands are grouped
according to energy per photon we obtain the spectrum shown in the
image above ranging from gamma-rays (highest energy) to radio
waves (lowest energy)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Bibliography
bull RC Gonzales RE Woods Digital Image Processing Prentice Hall2008 3rd ed
bull RC Gonzales RE Woods SL Eddins Digital Image Processing Using MATLAB Prentice Hall 2003
bull httpwwwimageprocessingplacecom
bullM Petrou C Petrou Image Processing the fundamentals John Wiley 2010 2nd ed
bull W Burger MJ Burge Digital Image Processing An Algorithmic Introduction Using Java Springer 2008
Digital Image ProcessingDigital Image Processing
Week 1Week 1
bull Image Processing Toolbox (httpwwwmathworkscomproductsimage)
bull C Solomon T Breckon Fundamentals of Digital Image Processing A Practical Approach with Examples in Matlab Wiley-Blackwell 2011
bullWK Pratt Digital Image Processing Wiley-Interscience 2007
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Meet LenaThe First Lady of the Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lenna Soderberg (Sjoumloumlblom) and Jeff Seideman taken in May 1997Imaging Science amp Technology Conference
Digital Image ProcessingDigital Image Processing
Week 1Week 1
What is Digital Image Processing
f(xy) = intensity gray level of the image at spatial point (xy)
x y f(xy) ndash finite discrete quantities -gt digital image
Digital Image Processing = processing digital images by means of a digital computer
A digital image is composed of a finite number of elements (location value of intensity)
These elements are called picture elements image elements pels pixels
( )i j ijx y f
3 f D
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image processing is not limited to the visual band of the electromagnetic (EM) spectrum
Image processing gamma to radio waves ultrasound electron microscopy computer-generated images
image processing image analysis computer vision
Image processing = discipline in which both the input and the output of a process are images
Computer Vision = use computer to emulate human vision (AI) ndashlearning making inferences and take actions based on visual inputs
Image analysis (image understanding) = segmentation partitioning images into regions or objects
(link between image processing and image analysis)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Distinction between image processing image analysis computer vision
low-level mid-level high-level processes
Low-level processes image preprocessing to reduce noise contrast enhancement image sharpening both input and output are images
Mid-level processes segmentation partitioning images into regions or objects description of the objects for computer processing classificationrecognition of individual objects inputs are generally images outputs are attributes extracted from the input image (eg edges contours identity of individual objects)
High-level processes ldquomaking senserdquo of a set of recognized objects performing the cognitive functions associated with vision
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing (Gonzalez + Woods) =
processes whose inputs and outputs are images +
processes that extract attributes from images recognition of individual objects
(low- and mid-level processes)
Example
automated analysis of text = acquiring an image containing text preprocessing the image (enhancement sharpening) extracting (segmenting) the individual characaters describing the characters in a form suitable for computer processing recognition of individual characters
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The Origins of DIP
Newspaper industry pictures were sent by submarine cable between London and New York
Before Bartlane cable picture transmission system (early 1920s) ndash 1 week
With Bartlane system less than 3 hours
Specialized printing equipment coded pictures for cable transmission and reconstructed them at the receiving end (1920s -5 distict levels of gray 1929 ndash 15 levels)
This example is not DIP the computer is not involved
DIP is linked and devolps in the same rhythm as digital computers (data storage display and transmission)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
A digital pictureproduced in 1921from a coded tapeby a telegraph printer withspecial type faces(McFarlane)
A digital picturemade in 1922from a tapepunched after the signals hadcrossed the Atlantic twice(McFarlane)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1964 Jet Propulsion Laboratory (Pasadena California) processed pictures of the moon transmitted by Ranger 7 (corrected image distortions)
The first picture of the moon by a US spacecraft Ranger 7 took this image July 31 1964 about 17 minutes before impacting the lunar surface(Courtesy of NASA)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1960-1970 ndash image processing techniques were used in medical image remote Earth resources observations astronomy
1970s ndash invention of CAT (computerized axial tomography)
CAT is a process in which a ring of detectors encircles an object (patient) and a X-ray source concentric with the detector ring rotates about the object The X-ray passes through the patient and the resulted waves are collected at the opposite end by the detectors As the source rotates the procedure is repeated Tomography consists of algorithms that use the sense data to construct an image that represent a ldquoslicerdquo through the object Motion of the object in a direction perpendicular to the ring of detectors produces a set of ldquoslicesrdquo which can be assembled in a 3D information of the inside of the object
Digital Image ProcessingDigital Image Processing
Week 1Week 1
loz geographers use DIP to study pollution patterns from aerial and satellite imagery
loz archeology ndash DIP allowed restoring blurred pictures that recorded rare artifacts lost or damaged after being photographed
loz physics ndash enhance images of experiments (high-energy plasmas electron microscopy)
loz astronomy biology nuclear medicine law enforcement industry
DIP ndash used in solving problems dealing with machine perception ndashextracting from an image information suitable for computer processing (statistical moments Fourier transform coefficients hellip)
loz automatic character recognition industrial machine vision for product assembly and inspection military recognizance automatic processing of fingerprints machine processing of aerial and satellite imagery for weather prediction Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of Fields that Use DIP
Images can be classified according to their sources (visual X-ray hellip)
Energy sources for images electromagnetic energy spectrum acoustic ultrasonic electronic computer- generated
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Electromagnetic waves can be thought as propagating sinusoidal
waves of different wavelength or as a stream of massless particles
each moving in a wavelike pattern with the speed of light Each
massless particle contains a certain amount (bundle) of energy Each
bundle of energy is called a photon If spectral bands are grouped
according to energy per photon we obtain the spectrum shown in the
image above ranging from gamma-rays (highest energy) to radio
waves (lowest energy)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
bull Image Processing Toolbox (httpwwwmathworkscomproductsimage)
bull C Solomon T Breckon Fundamentals of Digital Image Processing A Practical Approach with Examples in Matlab Wiley-Blackwell 2011
bullWK Pratt Digital Image Processing Wiley-Interscience 2007
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Meet LenaThe First Lady of the Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lenna Soderberg (Sjoumloumlblom) and Jeff Seideman taken in May 1997Imaging Science amp Technology Conference
Digital Image ProcessingDigital Image Processing
Week 1Week 1
What is Digital Image Processing
f(xy) = intensity gray level of the image at spatial point (xy)
x y f(xy) ndash finite discrete quantities -gt digital image
Digital Image Processing = processing digital images by means of a digital computer
A digital image is composed of a finite number of elements (location value of intensity)
These elements are called picture elements image elements pels pixels
( )i j ijx y f
3 f D
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image processing is not limited to the visual band of the electromagnetic (EM) spectrum
Image processing gamma to radio waves ultrasound electron microscopy computer-generated images
image processing image analysis computer vision
Image processing = discipline in which both the input and the output of a process are images
Computer Vision = use computer to emulate human vision (AI) ndashlearning making inferences and take actions based on visual inputs
Image analysis (image understanding) = segmentation partitioning images into regions or objects
(link between image processing and image analysis)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Distinction between image processing image analysis computer vision
low-level mid-level high-level processes
Low-level processes image preprocessing to reduce noise contrast enhancement image sharpening both input and output are images
Mid-level processes segmentation partitioning images into regions or objects description of the objects for computer processing classificationrecognition of individual objects inputs are generally images outputs are attributes extracted from the input image (eg edges contours identity of individual objects)
High-level processes ldquomaking senserdquo of a set of recognized objects performing the cognitive functions associated with vision
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing (Gonzalez + Woods) =
processes whose inputs and outputs are images +
processes that extract attributes from images recognition of individual objects
(low- and mid-level processes)
Example
automated analysis of text = acquiring an image containing text preprocessing the image (enhancement sharpening) extracting (segmenting) the individual characaters describing the characters in a form suitable for computer processing recognition of individual characters
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The Origins of DIP
Newspaper industry pictures were sent by submarine cable between London and New York
Before Bartlane cable picture transmission system (early 1920s) ndash 1 week
With Bartlane system less than 3 hours
Specialized printing equipment coded pictures for cable transmission and reconstructed them at the receiving end (1920s -5 distict levels of gray 1929 ndash 15 levels)
This example is not DIP the computer is not involved
DIP is linked and devolps in the same rhythm as digital computers (data storage display and transmission)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
A digital pictureproduced in 1921from a coded tapeby a telegraph printer withspecial type faces(McFarlane)
A digital picturemade in 1922from a tapepunched after the signals hadcrossed the Atlantic twice(McFarlane)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1964 Jet Propulsion Laboratory (Pasadena California) processed pictures of the moon transmitted by Ranger 7 (corrected image distortions)
The first picture of the moon by a US spacecraft Ranger 7 took this image July 31 1964 about 17 minutes before impacting the lunar surface(Courtesy of NASA)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1960-1970 ndash image processing techniques were used in medical image remote Earth resources observations astronomy
1970s ndash invention of CAT (computerized axial tomography)
CAT is a process in which a ring of detectors encircles an object (patient) and a X-ray source concentric with the detector ring rotates about the object The X-ray passes through the patient and the resulted waves are collected at the opposite end by the detectors As the source rotates the procedure is repeated Tomography consists of algorithms that use the sense data to construct an image that represent a ldquoslicerdquo through the object Motion of the object in a direction perpendicular to the ring of detectors produces a set of ldquoslicesrdquo which can be assembled in a 3D information of the inside of the object
Digital Image ProcessingDigital Image Processing
Week 1Week 1
loz geographers use DIP to study pollution patterns from aerial and satellite imagery
loz archeology ndash DIP allowed restoring blurred pictures that recorded rare artifacts lost or damaged after being photographed
loz physics ndash enhance images of experiments (high-energy plasmas electron microscopy)
loz astronomy biology nuclear medicine law enforcement industry
DIP ndash used in solving problems dealing with machine perception ndashextracting from an image information suitable for computer processing (statistical moments Fourier transform coefficients hellip)
loz automatic character recognition industrial machine vision for product assembly and inspection military recognizance automatic processing of fingerprints machine processing of aerial and satellite imagery for weather prediction Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of Fields that Use DIP
Images can be classified according to their sources (visual X-ray hellip)
Energy sources for images electromagnetic energy spectrum acoustic ultrasonic electronic computer- generated
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Electromagnetic waves can be thought as propagating sinusoidal
waves of different wavelength or as a stream of massless particles
each moving in a wavelike pattern with the speed of light Each
massless particle contains a certain amount (bundle) of energy Each
bundle of energy is called a photon If spectral bands are grouped
according to energy per photon we obtain the spectrum shown in the
image above ranging from gamma-rays (highest energy) to radio
waves (lowest energy)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Meet LenaThe First Lady of the Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lenna Soderberg (Sjoumloumlblom) and Jeff Seideman taken in May 1997Imaging Science amp Technology Conference
Digital Image ProcessingDigital Image Processing
Week 1Week 1
What is Digital Image Processing
f(xy) = intensity gray level of the image at spatial point (xy)
x y f(xy) ndash finite discrete quantities -gt digital image
Digital Image Processing = processing digital images by means of a digital computer
A digital image is composed of a finite number of elements (location value of intensity)
These elements are called picture elements image elements pels pixels
( )i j ijx y f
3 f D
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image processing is not limited to the visual band of the electromagnetic (EM) spectrum
Image processing gamma to radio waves ultrasound electron microscopy computer-generated images
image processing image analysis computer vision
Image processing = discipline in which both the input and the output of a process are images
Computer Vision = use computer to emulate human vision (AI) ndashlearning making inferences and take actions based on visual inputs
Image analysis (image understanding) = segmentation partitioning images into regions or objects
(link between image processing and image analysis)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Distinction between image processing image analysis computer vision
low-level mid-level high-level processes
Low-level processes image preprocessing to reduce noise contrast enhancement image sharpening both input and output are images
Mid-level processes segmentation partitioning images into regions or objects description of the objects for computer processing classificationrecognition of individual objects inputs are generally images outputs are attributes extracted from the input image (eg edges contours identity of individual objects)
High-level processes ldquomaking senserdquo of a set of recognized objects performing the cognitive functions associated with vision
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing (Gonzalez + Woods) =
processes whose inputs and outputs are images +
processes that extract attributes from images recognition of individual objects
(low- and mid-level processes)
Example
automated analysis of text = acquiring an image containing text preprocessing the image (enhancement sharpening) extracting (segmenting) the individual characaters describing the characters in a form suitable for computer processing recognition of individual characters
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The Origins of DIP
Newspaper industry pictures were sent by submarine cable between London and New York
Before Bartlane cable picture transmission system (early 1920s) ndash 1 week
With Bartlane system less than 3 hours
Specialized printing equipment coded pictures for cable transmission and reconstructed them at the receiving end (1920s -5 distict levels of gray 1929 ndash 15 levels)
This example is not DIP the computer is not involved
DIP is linked and devolps in the same rhythm as digital computers (data storage display and transmission)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
A digital pictureproduced in 1921from a coded tapeby a telegraph printer withspecial type faces(McFarlane)
A digital picturemade in 1922from a tapepunched after the signals hadcrossed the Atlantic twice(McFarlane)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1964 Jet Propulsion Laboratory (Pasadena California) processed pictures of the moon transmitted by Ranger 7 (corrected image distortions)
The first picture of the moon by a US spacecraft Ranger 7 took this image July 31 1964 about 17 minutes before impacting the lunar surface(Courtesy of NASA)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1960-1970 ndash image processing techniques were used in medical image remote Earth resources observations astronomy
1970s ndash invention of CAT (computerized axial tomography)
CAT is a process in which a ring of detectors encircles an object (patient) and a X-ray source concentric with the detector ring rotates about the object The X-ray passes through the patient and the resulted waves are collected at the opposite end by the detectors As the source rotates the procedure is repeated Tomography consists of algorithms that use the sense data to construct an image that represent a ldquoslicerdquo through the object Motion of the object in a direction perpendicular to the ring of detectors produces a set of ldquoslicesrdquo which can be assembled in a 3D information of the inside of the object
Digital Image ProcessingDigital Image Processing
Week 1Week 1
loz geographers use DIP to study pollution patterns from aerial and satellite imagery
loz archeology ndash DIP allowed restoring blurred pictures that recorded rare artifacts lost or damaged after being photographed
loz physics ndash enhance images of experiments (high-energy plasmas electron microscopy)
loz astronomy biology nuclear medicine law enforcement industry
DIP ndash used in solving problems dealing with machine perception ndashextracting from an image information suitable for computer processing (statistical moments Fourier transform coefficients hellip)
loz automatic character recognition industrial machine vision for product assembly and inspection military recognizance automatic processing of fingerprints machine processing of aerial and satellite imagery for weather prediction Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of Fields that Use DIP
Images can be classified according to their sources (visual X-ray hellip)
Energy sources for images electromagnetic energy spectrum acoustic ultrasonic electronic computer- generated
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Electromagnetic waves can be thought as propagating sinusoidal
waves of different wavelength or as a stream of massless particles
each moving in a wavelike pattern with the speed of light Each
massless particle contains a certain amount (bundle) of energy Each
bundle of energy is called a photon If spectral bands are grouped
according to energy per photon we obtain the spectrum shown in the
image above ranging from gamma-rays (highest energy) to radio
waves (lowest energy)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Meet LenaThe First Lady of the Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lenna Soderberg (Sjoumloumlblom) and Jeff Seideman taken in May 1997Imaging Science amp Technology Conference
Digital Image ProcessingDigital Image Processing
Week 1Week 1
What is Digital Image Processing
f(xy) = intensity gray level of the image at spatial point (xy)
x y f(xy) ndash finite discrete quantities -gt digital image
Digital Image Processing = processing digital images by means of a digital computer
A digital image is composed of a finite number of elements (location value of intensity)
These elements are called picture elements image elements pels pixels
( )i j ijx y f
3 f D
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image processing is not limited to the visual band of the electromagnetic (EM) spectrum
Image processing gamma to radio waves ultrasound electron microscopy computer-generated images
image processing image analysis computer vision
Image processing = discipline in which both the input and the output of a process are images
Computer Vision = use computer to emulate human vision (AI) ndashlearning making inferences and take actions based on visual inputs
Image analysis (image understanding) = segmentation partitioning images into regions or objects
(link between image processing and image analysis)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Distinction between image processing image analysis computer vision
low-level mid-level high-level processes
Low-level processes image preprocessing to reduce noise contrast enhancement image sharpening both input and output are images
Mid-level processes segmentation partitioning images into regions or objects description of the objects for computer processing classificationrecognition of individual objects inputs are generally images outputs are attributes extracted from the input image (eg edges contours identity of individual objects)
High-level processes ldquomaking senserdquo of a set of recognized objects performing the cognitive functions associated with vision
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing (Gonzalez + Woods) =
processes whose inputs and outputs are images +
processes that extract attributes from images recognition of individual objects
(low- and mid-level processes)
Example
automated analysis of text = acquiring an image containing text preprocessing the image (enhancement sharpening) extracting (segmenting) the individual characaters describing the characters in a form suitable for computer processing recognition of individual characters
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The Origins of DIP
Newspaper industry pictures were sent by submarine cable between London and New York
Before Bartlane cable picture transmission system (early 1920s) ndash 1 week
With Bartlane system less than 3 hours
Specialized printing equipment coded pictures for cable transmission and reconstructed them at the receiving end (1920s -5 distict levels of gray 1929 ndash 15 levels)
This example is not DIP the computer is not involved
DIP is linked and devolps in the same rhythm as digital computers (data storage display and transmission)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
A digital pictureproduced in 1921from a coded tapeby a telegraph printer withspecial type faces(McFarlane)
A digital picturemade in 1922from a tapepunched after the signals hadcrossed the Atlantic twice(McFarlane)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1964 Jet Propulsion Laboratory (Pasadena California) processed pictures of the moon transmitted by Ranger 7 (corrected image distortions)
The first picture of the moon by a US spacecraft Ranger 7 took this image July 31 1964 about 17 minutes before impacting the lunar surface(Courtesy of NASA)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1960-1970 ndash image processing techniques were used in medical image remote Earth resources observations astronomy
1970s ndash invention of CAT (computerized axial tomography)
CAT is a process in which a ring of detectors encircles an object (patient) and a X-ray source concentric with the detector ring rotates about the object The X-ray passes through the patient and the resulted waves are collected at the opposite end by the detectors As the source rotates the procedure is repeated Tomography consists of algorithms that use the sense data to construct an image that represent a ldquoslicerdquo through the object Motion of the object in a direction perpendicular to the ring of detectors produces a set of ldquoslicesrdquo which can be assembled in a 3D information of the inside of the object
Digital Image ProcessingDigital Image Processing
Week 1Week 1
loz geographers use DIP to study pollution patterns from aerial and satellite imagery
loz archeology ndash DIP allowed restoring blurred pictures that recorded rare artifacts lost or damaged after being photographed
loz physics ndash enhance images of experiments (high-energy plasmas electron microscopy)
loz astronomy biology nuclear medicine law enforcement industry
DIP ndash used in solving problems dealing with machine perception ndashextracting from an image information suitable for computer processing (statistical moments Fourier transform coefficients hellip)
loz automatic character recognition industrial machine vision for product assembly and inspection military recognizance automatic processing of fingerprints machine processing of aerial and satellite imagery for weather prediction Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of Fields that Use DIP
Images can be classified according to their sources (visual X-ray hellip)
Energy sources for images electromagnetic energy spectrum acoustic ultrasonic electronic computer- generated
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Electromagnetic waves can be thought as propagating sinusoidal
waves of different wavelength or as a stream of massless particles
each moving in a wavelike pattern with the speed of light Each
massless particle contains a certain amount (bundle) of energy Each
bundle of energy is called a photon If spectral bands are grouped
according to energy per photon we obtain the spectrum shown in the
image above ranging from gamma-rays (highest energy) to radio
waves (lowest energy)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lenna Soderberg (Sjoumloumlblom) and Jeff Seideman taken in May 1997Imaging Science amp Technology Conference
Digital Image ProcessingDigital Image Processing
Week 1Week 1
What is Digital Image Processing
f(xy) = intensity gray level of the image at spatial point (xy)
x y f(xy) ndash finite discrete quantities -gt digital image
Digital Image Processing = processing digital images by means of a digital computer
A digital image is composed of a finite number of elements (location value of intensity)
These elements are called picture elements image elements pels pixels
( )i j ijx y f
3 f D
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image processing is not limited to the visual band of the electromagnetic (EM) spectrum
Image processing gamma to radio waves ultrasound electron microscopy computer-generated images
image processing image analysis computer vision
Image processing = discipline in which both the input and the output of a process are images
Computer Vision = use computer to emulate human vision (AI) ndashlearning making inferences and take actions based on visual inputs
Image analysis (image understanding) = segmentation partitioning images into regions or objects
(link between image processing and image analysis)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Distinction between image processing image analysis computer vision
low-level mid-level high-level processes
Low-level processes image preprocessing to reduce noise contrast enhancement image sharpening both input and output are images
Mid-level processes segmentation partitioning images into regions or objects description of the objects for computer processing classificationrecognition of individual objects inputs are generally images outputs are attributes extracted from the input image (eg edges contours identity of individual objects)
High-level processes ldquomaking senserdquo of a set of recognized objects performing the cognitive functions associated with vision
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing (Gonzalez + Woods) =
processes whose inputs and outputs are images +
processes that extract attributes from images recognition of individual objects
(low- and mid-level processes)
Example
automated analysis of text = acquiring an image containing text preprocessing the image (enhancement sharpening) extracting (segmenting) the individual characaters describing the characters in a form suitable for computer processing recognition of individual characters
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The Origins of DIP
Newspaper industry pictures were sent by submarine cable between London and New York
Before Bartlane cable picture transmission system (early 1920s) ndash 1 week
With Bartlane system less than 3 hours
Specialized printing equipment coded pictures for cable transmission and reconstructed them at the receiving end (1920s -5 distict levels of gray 1929 ndash 15 levels)
This example is not DIP the computer is not involved
DIP is linked and devolps in the same rhythm as digital computers (data storage display and transmission)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
A digital pictureproduced in 1921from a coded tapeby a telegraph printer withspecial type faces(McFarlane)
A digital picturemade in 1922from a tapepunched after the signals hadcrossed the Atlantic twice(McFarlane)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1964 Jet Propulsion Laboratory (Pasadena California) processed pictures of the moon transmitted by Ranger 7 (corrected image distortions)
The first picture of the moon by a US spacecraft Ranger 7 took this image July 31 1964 about 17 minutes before impacting the lunar surface(Courtesy of NASA)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1960-1970 ndash image processing techniques were used in medical image remote Earth resources observations astronomy
1970s ndash invention of CAT (computerized axial tomography)
CAT is a process in which a ring of detectors encircles an object (patient) and a X-ray source concentric with the detector ring rotates about the object The X-ray passes through the patient and the resulted waves are collected at the opposite end by the detectors As the source rotates the procedure is repeated Tomography consists of algorithms that use the sense data to construct an image that represent a ldquoslicerdquo through the object Motion of the object in a direction perpendicular to the ring of detectors produces a set of ldquoslicesrdquo which can be assembled in a 3D information of the inside of the object
Digital Image ProcessingDigital Image Processing
Week 1Week 1
loz geographers use DIP to study pollution patterns from aerial and satellite imagery
loz archeology ndash DIP allowed restoring blurred pictures that recorded rare artifacts lost or damaged after being photographed
loz physics ndash enhance images of experiments (high-energy plasmas electron microscopy)
loz astronomy biology nuclear medicine law enforcement industry
DIP ndash used in solving problems dealing with machine perception ndashextracting from an image information suitable for computer processing (statistical moments Fourier transform coefficients hellip)
loz automatic character recognition industrial machine vision for product assembly and inspection military recognizance automatic processing of fingerprints machine processing of aerial and satellite imagery for weather prediction Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of Fields that Use DIP
Images can be classified according to their sources (visual X-ray hellip)
Energy sources for images electromagnetic energy spectrum acoustic ultrasonic electronic computer- generated
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Electromagnetic waves can be thought as propagating sinusoidal
waves of different wavelength or as a stream of massless particles
each moving in a wavelike pattern with the speed of light Each
massless particle contains a certain amount (bundle) of energy Each
bundle of energy is called a photon If spectral bands are grouped
according to energy per photon we obtain the spectrum shown in the
image above ranging from gamma-rays (highest energy) to radio
waves (lowest energy)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
What is Digital Image Processing
f(xy) = intensity gray level of the image at spatial point (xy)
x y f(xy) ndash finite discrete quantities -gt digital image
Digital Image Processing = processing digital images by means of a digital computer
A digital image is composed of a finite number of elements (location value of intensity)
These elements are called picture elements image elements pels pixels
( )i j ijx y f
3 f D
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image processing is not limited to the visual band of the electromagnetic (EM) spectrum
Image processing gamma to radio waves ultrasound electron microscopy computer-generated images
image processing image analysis computer vision
Image processing = discipline in which both the input and the output of a process are images
Computer Vision = use computer to emulate human vision (AI) ndashlearning making inferences and take actions based on visual inputs
Image analysis (image understanding) = segmentation partitioning images into regions or objects
(link between image processing and image analysis)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Distinction between image processing image analysis computer vision
low-level mid-level high-level processes
Low-level processes image preprocessing to reduce noise contrast enhancement image sharpening both input and output are images
Mid-level processes segmentation partitioning images into regions or objects description of the objects for computer processing classificationrecognition of individual objects inputs are generally images outputs are attributes extracted from the input image (eg edges contours identity of individual objects)
High-level processes ldquomaking senserdquo of a set of recognized objects performing the cognitive functions associated with vision
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing (Gonzalez + Woods) =
processes whose inputs and outputs are images +
processes that extract attributes from images recognition of individual objects
(low- and mid-level processes)
Example
automated analysis of text = acquiring an image containing text preprocessing the image (enhancement sharpening) extracting (segmenting) the individual characaters describing the characters in a form suitable for computer processing recognition of individual characters
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The Origins of DIP
Newspaper industry pictures were sent by submarine cable between London and New York
Before Bartlane cable picture transmission system (early 1920s) ndash 1 week
With Bartlane system less than 3 hours
Specialized printing equipment coded pictures for cable transmission and reconstructed them at the receiving end (1920s -5 distict levels of gray 1929 ndash 15 levels)
This example is not DIP the computer is not involved
DIP is linked and devolps in the same rhythm as digital computers (data storage display and transmission)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
A digital pictureproduced in 1921from a coded tapeby a telegraph printer withspecial type faces(McFarlane)
A digital picturemade in 1922from a tapepunched after the signals hadcrossed the Atlantic twice(McFarlane)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1964 Jet Propulsion Laboratory (Pasadena California) processed pictures of the moon transmitted by Ranger 7 (corrected image distortions)
The first picture of the moon by a US spacecraft Ranger 7 took this image July 31 1964 about 17 minutes before impacting the lunar surface(Courtesy of NASA)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1960-1970 ndash image processing techniques were used in medical image remote Earth resources observations astronomy
1970s ndash invention of CAT (computerized axial tomography)
CAT is a process in which a ring of detectors encircles an object (patient) and a X-ray source concentric with the detector ring rotates about the object The X-ray passes through the patient and the resulted waves are collected at the opposite end by the detectors As the source rotates the procedure is repeated Tomography consists of algorithms that use the sense data to construct an image that represent a ldquoslicerdquo through the object Motion of the object in a direction perpendicular to the ring of detectors produces a set of ldquoslicesrdquo which can be assembled in a 3D information of the inside of the object
Digital Image ProcessingDigital Image Processing
Week 1Week 1
loz geographers use DIP to study pollution patterns from aerial and satellite imagery
loz archeology ndash DIP allowed restoring blurred pictures that recorded rare artifacts lost or damaged after being photographed
loz physics ndash enhance images of experiments (high-energy plasmas electron microscopy)
loz astronomy biology nuclear medicine law enforcement industry
DIP ndash used in solving problems dealing with machine perception ndashextracting from an image information suitable for computer processing (statistical moments Fourier transform coefficients hellip)
loz automatic character recognition industrial machine vision for product assembly and inspection military recognizance automatic processing of fingerprints machine processing of aerial and satellite imagery for weather prediction Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of Fields that Use DIP
Images can be classified according to their sources (visual X-ray hellip)
Energy sources for images electromagnetic energy spectrum acoustic ultrasonic electronic computer- generated
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Electromagnetic waves can be thought as propagating sinusoidal
waves of different wavelength or as a stream of massless particles
each moving in a wavelike pattern with the speed of light Each
massless particle contains a certain amount (bundle) of energy Each
bundle of energy is called a photon If spectral bands are grouped
according to energy per photon we obtain the spectrum shown in the
image above ranging from gamma-rays (highest energy) to radio
waves (lowest energy)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image processing is not limited to the visual band of the electromagnetic (EM) spectrum
Image processing gamma to radio waves ultrasound electron microscopy computer-generated images
image processing image analysis computer vision
Image processing = discipline in which both the input and the output of a process are images
Computer Vision = use computer to emulate human vision (AI) ndashlearning making inferences and take actions based on visual inputs
Image analysis (image understanding) = segmentation partitioning images into regions or objects
(link between image processing and image analysis)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Distinction between image processing image analysis computer vision
low-level mid-level high-level processes
Low-level processes image preprocessing to reduce noise contrast enhancement image sharpening both input and output are images
Mid-level processes segmentation partitioning images into regions or objects description of the objects for computer processing classificationrecognition of individual objects inputs are generally images outputs are attributes extracted from the input image (eg edges contours identity of individual objects)
High-level processes ldquomaking senserdquo of a set of recognized objects performing the cognitive functions associated with vision
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing (Gonzalez + Woods) =
processes whose inputs and outputs are images +
processes that extract attributes from images recognition of individual objects
(low- and mid-level processes)
Example
automated analysis of text = acquiring an image containing text preprocessing the image (enhancement sharpening) extracting (segmenting) the individual characaters describing the characters in a form suitable for computer processing recognition of individual characters
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The Origins of DIP
Newspaper industry pictures were sent by submarine cable between London and New York
Before Bartlane cable picture transmission system (early 1920s) ndash 1 week
With Bartlane system less than 3 hours
Specialized printing equipment coded pictures for cable transmission and reconstructed them at the receiving end (1920s -5 distict levels of gray 1929 ndash 15 levels)
This example is not DIP the computer is not involved
DIP is linked and devolps in the same rhythm as digital computers (data storage display and transmission)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
A digital pictureproduced in 1921from a coded tapeby a telegraph printer withspecial type faces(McFarlane)
A digital picturemade in 1922from a tapepunched after the signals hadcrossed the Atlantic twice(McFarlane)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1964 Jet Propulsion Laboratory (Pasadena California) processed pictures of the moon transmitted by Ranger 7 (corrected image distortions)
The first picture of the moon by a US spacecraft Ranger 7 took this image July 31 1964 about 17 minutes before impacting the lunar surface(Courtesy of NASA)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1960-1970 ndash image processing techniques were used in medical image remote Earth resources observations astronomy
1970s ndash invention of CAT (computerized axial tomography)
CAT is a process in which a ring of detectors encircles an object (patient) and a X-ray source concentric with the detector ring rotates about the object The X-ray passes through the patient and the resulted waves are collected at the opposite end by the detectors As the source rotates the procedure is repeated Tomography consists of algorithms that use the sense data to construct an image that represent a ldquoslicerdquo through the object Motion of the object in a direction perpendicular to the ring of detectors produces a set of ldquoslicesrdquo which can be assembled in a 3D information of the inside of the object
Digital Image ProcessingDigital Image Processing
Week 1Week 1
loz geographers use DIP to study pollution patterns from aerial and satellite imagery
loz archeology ndash DIP allowed restoring blurred pictures that recorded rare artifacts lost or damaged after being photographed
loz physics ndash enhance images of experiments (high-energy plasmas electron microscopy)
loz astronomy biology nuclear medicine law enforcement industry
DIP ndash used in solving problems dealing with machine perception ndashextracting from an image information suitable for computer processing (statistical moments Fourier transform coefficients hellip)
loz automatic character recognition industrial machine vision for product assembly and inspection military recognizance automatic processing of fingerprints machine processing of aerial and satellite imagery for weather prediction Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of Fields that Use DIP
Images can be classified according to their sources (visual X-ray hellip)
Energy sources for images electromagnetic energy spectrum acoustic ultrasonic electronic computer- generated
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Electromagnetic waves can be thought as propagating sinusoidal
waves of different wavelength or as a stream of massless particles
each moving in a wavelike pattern with the speed of light Each
massless particle contains a certain amount (bundle) of energy Each
bundle of energy is called a photon If spectral bands are grouped
according to energy per photon we obtain the spectrum shown in the
image above ranging from gamma-rays (highest energy) to radio
waves (lowest energy)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Distinction between image processing image analysis computer vision
low-level mid-level high-level processes
Low-level processes image preprocessing to reduce noise contrast enhancement image sharpening both input and output are images
Mid-level processes segmentation partitioning images into regions or objects description of the objects for computer processing classificationrecognition of individual objects inputs are generally images outputs are attributes extracted from the input image (eg edges contours identity of individual objects)
High-level processes ldquomaking senserdquo of a set of recognized objects performing the cognitive functions associated with vision
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing (Gonzalez + Woods) =
processes whose inputs and outputs are images +
processes that extract attributes from images recognition of individual objects
(low- and mid-level processes)
Example
automated analysis of text = acquiring an image containing text preprocessing the image (enhancement sharpening) extracting (segmenting) the individual characaters describing the characters in a form suitable for computer processing recognition of individual characters
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The Origins of DIP
Newspaper industry pictures were sent by submarine cable between London and New York
Before Bartlane cable picture transmission system (early 1920s) ndash 1 week
With Bartlane system less than 3 hours
Specialized printing equipment coded pictures for cable transmission and reconstructed them at the receiving end (1920s -5 distict levels of gray 1929 ndash 15 levels)
This example is not DIP the computer is not involved
DIP is linked and devolps in the same rhythm as digital computers (data storage display and transmission)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
A digital pictureproduced in 1921from a coded tapeby a telegraph printer withspecial type faces(McFarlane)
A digital picturemade in 1922from a tapepunched after the signals hadcrossed the Atlantic twice(McFarlane)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1964 Jet Propulsion Laboratory (Pasadena California) processed pictures of the moon transmitted by Ranger 7 (corrected image distortions)
The first picture of the moon by a US spacecraft Ranger 7 took this image July 31 1964 about 17 minutes before impacting the lunar surface(Courtesy of NASA)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1960-1970 ndash image processing techniques were used in medical image remote Earth resources observations astronomy
1970s ndash invention of CAT (computerized axial tomography)
CAT is a process in which a ring of detectors encircles an object (patient) and a X-ray source concentric with the detector ring rotates about the object The X-ray passes through the patient and the resulted waves are collected at the opposite end by the detectors As the source rotates the procedure is repeated Tomography consists of algorithms that use the sense data to construct an image that represent a ldquoslicerdquo through the object Motion of the object in a direction perpendicular to the ring of detectors produces a set of ldquoslicesrdquo which can be assembled in a 3D information of the inside of the object
Digital Image ProcessingDigital Image Processing
Week 1Week 1
loz geographers use DIP to study pollution patterns from aerial and satellite imagery
loz archeology ndash DIP allowed restoring blurred pictures that recorded rare artifacts lost or damaged after being photographed
loz physics ndash enhance images of experiments (high-energy plasmas electron microscopy)
loz astronomy biology nuclear medicine law enforcement industry
DIP ndash used in solving problems dealing with machine perception ndashextracting from an image information suitable for computer processing (statistical moments Fourier transform coefficients hellip)
loz automatic character recognition industrial machine vision for product assembly and inspection military recognizance automatic processing of fingerprints machine processing of aerial and satellite imagery for weather prediction Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of Fields that Use DIP
Images can be classified according to their sources (visual X-ray hellip)
Energy sources for images electromagnetic energy spectrum acoustic ultrasonic electronic computer- generated
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Electromagnetic waves can be thought as propagating sinusoidal
waves of different wavelength or as a stream of massless particles
each moving in a wavelike pattern with the speed of light Each
massless particle contains a certain amount (bundle) of energy Each
bundle of energy is called a photon If spectral bands are grouped
according to energy per photon we obtain the spectrum shown in the
image above ranging from gamma-rays (highest energy) to radio
waves (lowest energy)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing (Gonzalez + Woods) =
processes whose inputs and outputs are images +
processes that extract attributes from images recognition of individual objects
(low- and mid-level processes)
Example
automated analysis of text = acquiring an image containing text preprocessing the image (enhancement sharpening) extracting (segmenting) the individual characaters describing the characters in a form suitable for computer processing recognition of individual characters
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The Origins of DIP
Newspaper industry pictures were sent by submarine cable between London and New York
Before Bartlane cable picture transmission system (early 1920s) ndash 1 week
With Bartlane system less than 3 hours
Specialized printing equipment coded pictures for cable transmission and reconstructed them at the receiving end (1920s -5 distict levels of gray 1929 ndash 15 levels)
This example is not DIP the computer is not involved
DIP is linked and devolps in the same rhythm as digital computers (data storage display and transmission)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
A digital pictureproduced in 1921from a coded tapeby a telegraph printer withspecial type faces(McFarlane)
A digital picturemade in 1922from a tapepunched after the signals hadcrossed the Atlantic twice(McFarlane)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1964 Jet Propulsion Laboratory (Pasadena California) processed pictures of the moon transmitted by Ranger 7 (corrected image distortions)
The first picture of the moon by a US spacecraft Ranger 7 took this image July 31 1964 about 17 minutes before impacting the lunar surface(Courtesy of NASA)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1960-1970 ndash image processing techniques were used in medical image remote Earth resources observations astronomy
1970s ndash invention of CAT (computerized axial tomography)
CAT is a process in which a ring of detectors encircles an object (patient) and a X-ray source concentric with the detector ring rotates about the object The X-ray passes through the patient and the resulted waves are collected at the opposite end by the detectors As the source rotates the procedure is repeated Tomography consists of algorithms that use the sense data to construct an image that represent a ldquoslicerdquo through the object Motion of the object in a direction perpendicular to the ring of detectors produces a set of ldquoslicesrdquo which can be assembled in a 3D information of the inside of the object
Digital Image ProcessingDigital Image Processing
Week 1Week 1
loz geographers use DIP to study pollution patterns from aerial and satellite imagery
loz archeology ndash DIP allowed restoring blurred pictures that recorded rare artifacts lost or damaged after being photographed
loz physics ndash enhance images of experiments (high-energy plasmas electron microscopy)
loz astronomy biology nuclear medicine law enforcement industry
DIP ndash used in solving problems dealing with machine perception ndashextracting from an image information suitable for computer processing (statistical moments Fourier transform coefficients hellip)
loz automatic character recognition industrial machine vision for product assembly and inspection military recognizance automatic processing of fingerprints machine processing of aerial and satellite imagery for weather prediction Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of Fields that Use DIP
Images can be classified according to their sources (visual X-ray hellip)
Energy sources for images electromagnetic energy spectrum acoustic ultrasonic electronic computer- generated
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Electromagnetic waves can be thought as propagating sinusoidal
waves of different wavelength or as a stream of massless particles
each moving in a wavelike pattern with the speed of light Each
massless particle contains a certain amount (bundle) of energy Each
bundle of energy is called a photon If spectral bands are grouped
according to energy per photon we obtain the spectrum shown in the
image above ranging from gamma-rays (highest energy) to radio
waves (lowest energy)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The Origins of DIP
Newspaper industry pictures were sent by submarine cable between London and New York
Before Bartlane cable picture transmission system (early 1920s) ndash 1 week
With Bartlane system less than 3 hours
Specialized printing equipment coded pictures for cable transmission and reconstructed them at the receiving end (1920s -5 distict levels of gray 1929 ndash 15 levels)
This example is not DIP the computer is not involved
DIP is linked and devolps in the same rhythm as digital computers (data storage display and transmission)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
A digital pictureproduced in 1921from a coded tapeby a telegraph printer withspecial type faces(McFarlane)
A digital picturemade in 1922from a tapepunched after the signals hadcrossed the Atlantic twice(McFarlane)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1964 Jet Propulsion Laboratory (Pasadena California) processed pictures of the moon transmitted by Ranger 7 (corrected image distortions)
The first picture of the moon by a US spacecraft Ranger 7 took this image July 31 1964 about 17 minutes before impacting the lunar surface(Courtesy of NASA)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1960-1970 ndash image processing techniques were used in medical image remote Earth resources observations astronomy
1970s ndash invention of CAT (computerized axial tomography)
CAT is a process in which a ring of detectors encircles an object (patient) and a X-ray source concentric with the detector ring rotates about the object The X-ray passes through the patient and the resulted waves are collected at the opposite end by the detectors As the source rotates the procedure is repeated Tomography consists of algorithms that use the sense data to construct an image that represent a ldquoslicerdquo through the object Motion of the object in a direction perpendicular to the ring of detectors produces a set of ldquoslicesrdquo which can be assembled in a 3D information of the inside of the object
Digital Image ProcessingDigital Image Processing
Week 1Week 1
loz geographers use DIP to study pollution patterns from aerial and satellite imagery
loz archeology ndash DIP allowed restoring blurred pictures that recorded rare artifacts lost or damaged after being photographed
loz physics ndash enhance images of experiments (high-energy plasmas electron microscopy)
loz astronomy biology nuclear medicine law enforcement industry
DIP ndash used in solving problems dealing with machine perception ndashextracting from an image information suitable for computer processing (statistical moments Fourier transform coefficients hellip)
loz automatic character recognition industrial machine vision for product assembly and inspection military recognizance automatic processing of fingerprints machine processing of aerial and satellite imagery for weather prediction Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of Fields that Use DIP
Images can be classified according to their sources (visual X-ray hellip)
Energy sources for images electromagnetic energy spectrum acoustic ultrasonic electronic computer- generated
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Electromagnetic waves can be thought as propagating sinusoidal
waves of different wavelength or as a stream of massless particles
each moving in a wavelike pattern with the speed of light Each
massless particle contains a certain amount (bundle) of energy Each
bundle of energy is called a photon If spectral bands are grouped
according to energy per photon we obtain the spectrum shown in the
image above ranging from gamma-rays (highest energy) to radio
waves (lowest energy)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
A digital pictureproduced in 1921from a coded tapeby a telegraph printer withspecial type faces(McFarlane)
A digital picturemade in 1922from a tapepunched after the signals hadcrossed the Atlantic twice(McFarlane)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1964 Jet Propulsion Laboratory (Pasadena California) processed pictures of the moon transmitted by Ranger 7 (corrected image distortions)
The first picture of the moon by a US spacecraft Ranger 7 took this image July 31 1964 about 17 minutes before impacting the lunar surface(Courtesy of NASA)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1960-1970 ndash image processing techniques were used in medical image remote Earth resources observations astronomy
1970s ndash invention of CAT (computerized axial tomography)
CAT is a process in which a ring of detectors encircles an object (patient) and a X-ray source concentric with the detector ring rotates about the object The X-ray passes through the patient and the resulted waves are collected at the opposite end by the detectors As the source rotates the procedure is repeated Tomography consists of algorithms that use the sense data to construct an image that represent a ldquoslicerdquo through the object Motion of the object in a direction perpendicular to the ring of detectors produces a set of ldquoslicesrdquo which can be assembled in a 3D information of the inside of the object
Digital Image ProcessingDigital Image Processing
Week 1Week 1
loz geographers use DIP to study pollution patterns from aerial and satellite imagery
loz archeology ndash DIP allowed restoring blurred pictures that recorded rare artifacts lost or damaged after being photographed
loz physics ndash enhance images of experiments (high-energy plasmas electron microscopy)
loz astronomy biology nuclear medicine law enforcement industry
DIP ndash used in solving problems dealing with machine perception ndashextracting from an image information suitable for computer processing (statistical moments Fourier transform coefficients hellip)
loz automatic character recognition industrial machine vision for product assembly and inspection military recognizance automatic processing of fingerprints machine processing of aerial and satellite imagery for weather prediction Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of Fields that Use DIP
Images can be classified according to their sources (visual X-ray hellip)
Energy sources for images electromagnetic energy spectrum acoustic ultrasonic electronic computer- generated
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Electromagnetic waves can be thought as propagating sinusoidal
waves of different wavelength or as a stream of massless particles
each moving in a wavelike pattern with the speed of light Each
massless particle contains a certain amount (bundle) of energy Each
bundle of energy is called a photon If spectral bands are grouped
according to energy per photon we obtain the spectrum shown in the
image above ranging from gamma-rays (highest energy) to radio
waves (lowest energy)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1964 Jet Propulsion Laboratory (Pasadena California) processed pictures of the moon transmitted by Ranger 7 (corrected image distortions)
The first picture of the moon by a US spacecraft Ranger 7 took this image July 31 1964 about 17 minutes before impacting the lunar surface(Courtesy of NASA)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1960-1970 ndash image processing techniques were used in medical image remote Earth resources observations astronomy
1970s ndash invention of CAT (computerized axial tomography)
CAT is a process in which a ring of detectors encircles an object (patient) and a X-ray source concentric with the detector ring rotates about the object The X-ray passes through the patient and the resulted waves are collected at the opposite end by the detectors As the source rotates the procedure is repeated Tomography consists of algorithms that use the sense data to construct an image that represent a ldquoslicerdquo through the object Motion of the object in a direction perpendicular to the ring of detectors produces a set of ldquoslicesrdquo which can be assembled in a 3D information of the inside of the object
Digital Image ProcessingDigital Image Processing
Week 1Week 1
loz geographers use DIP to study pollution patterns from aerial and satellite imagery
loz archeology ndash DIP allowed restoring blurred pictures that recorded rare artifacts lost or damaged after being photographed
loz physics ndash enhance images of experiments (high-energy plasmas electron microscopy)
loz astronomy biology nuclear medicine law enforcement industry
DIP ndash used in solving problems dealing with machine perception ndashextracting from an image information suitable for computer processing (statistical moments Fourier transform coefficients hellip)
loz automatic character recognition industrial machine vision for product assembly and inspection military recognizance automatic processing of fingerprints machine processing of aerial and satellite imagery for weather prediction Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of Fields that Use DIP
Images can be classified according to their sources (visual X-ray hellip)
Energy sources for images electromagnetic energy spectrum acoustic ultrasonic electronic computer- generated
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Electromagnetic waves can be thought as propagating sinusoidal
waves of different wavelength or as a stream of massless particles
each moving in a wavelike pattern with the speed of light Each
massless particle contains a certain amount (bundle) of energy Each
bundle of energy is called a photon If spectral bands are grouped
according to energy per photon we obtain the spectrum shown in the
image above ranging from gamma-rays (highest energy) to radio
waves (lowest energy)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
1960-1970 ndash image processing techniques were used in medical image remote Earth resources observations astronomy
1970s ndash invention of CAT (computerized axial tomography)
CAT is a process in which a ring of detectors encircles an object (patient) and a X-ray source concentric with the detector ring rotates about the object The X-ray passes through the patient and the resulted waves are collected at the opposite end by the detectors As the source rotates the procedure is repeated Tomography consists of algorithms that use the sense data to construct an image that represent a ldquoslicerdquo through the object Motion of the object in a direction perpendicular to the ring of detectors produces a set of ldquoslicesrdquo which can be assembled in a 3D information of the inside of the object
Digital Image ProcessingDigital Image Processing
Week 1Week 1
loz geographers use DIP to study pollution patterns from aerial and satellite imagery
loz archeology ndash DIP allowed restoring blurred pictures that recorded rare artifacts lost or damaged after being photographed
loz physics ndash enhance images of experiments (high-energy plasmas electron microscopy)
loz astronomy biology nuclear medicine law enforcement industry
DIP ndash used in solving problems dealing with machine perception ndashextracting from an image information suitable for computer processing (statistical moments Fourier transform coefficients hellip)
loz automatic character recognition industrial machine vision for product assembly and inspection military recognizance automatic processing of fingerprints machine processing of aerial and satellite imagery for weather prediction Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of Fields that Use DIP
Images can be classified according to their sources (visual X-ray hellip)
Energy sources for images electromagnetic energy spectrum acoustic ultrasonic electronic computer- generated
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Electromagnetic waves can be thought as propagating sinusoidal
waves of different wavelength or as a stream of massless particles
each moving in a wavelike pattern with the speed of light Each
massless particle contains a certain amount (bundle) of energy Each
bundle of energy is called a photon If spectral bands are grouped
according to energy per photon we obtain the spectrum shown in the
image above ranging from gamma-rays (highest energy) to radio
waves (lowest energy)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
loz geographers use DIP to study pollution patterns from aerial and satellite imagery
loz archeology ndash DIP allowed restoring blurred pictures that recorded rare artifacts lost or damaged after being photographed
loz physics ndash enhance images of experiments (high-energy plasmas electron microscopy)
loz astronomy biology nuclear medicine law enforcement industry
DIP ndash used in solving problems dealing with machine perception ndashextracting from an image information suitable for computer processing (statistical moments Fourier transform coefficients hellip)
loz automatic character recognition industrial machine vision for product assembly and inspection military recognizance automatic processing of fingerprints machine processing of aerial and satellite imagery for weather prediction Internet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of Fields that Use DIP
Images can be classified according to their sources (visual X-ray hellip)
Energy sources for images electromagnetic energy spectrum acoustic ultrasonic electronic computer- generated
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Electromagnetic waves can be thought as propagating sinusoidal
waves of different wavelength or as a stream of massless particles
each moving in a wavelike pattern with the speed of light Each
massless particle contains a certain amount (bundle) of energy Each
bundle of energy is called a photon If spectral bands are grouped
according to energy per photon we obtain the spectrum shown in the
image above ranging from gamma-rays (highest energy) to radio
waves (lowest energy)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of Fields that Use DIP
Images can be classified according to their sources (visual X-ray hellip)
Energy sources for images electromagnetic energy spectrum acoustic ultrasonic electronic computer- generated
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Electromagnetic waves can be thought as propagating sinusoidal
waves of different wavelength or as a stream of massless particles
each moving in a wavelike pattern with the speed of light Each
massless particle contains a certain amount (bundle) of energy Each
bundle of energy is called a photon If spectral bands are grouped
according to energy per photon we obtain the spectrum shown in the
image above ranging from gamma-rays (highest energy) to radio
waves (lowest energy)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Electromagnetic waves can be thought as propagating sinusoidal
waves of different wavelength or as a stream of massless particles
each moving in a wavelike pattern with the speed of light Each
massless particle contains a certain amount (bundle) of energy Each
bundle of energy is called a photon If spectral bands are grouped
according to energy per photon we obtain the spectrum shown in the
image above ranging from gamma-rays (highest energy) to radio
waves (lowest energy)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gamma-Ray Imaging
Nuclear medicine astronomical observations
Nuclear medicine
the approach is to inject a patient with a radioactive isotope that emits gamma rays as it decays
Images are produced from the emissions collected by gamma-ray detectors
Images of this sort are used to locate sites of bone pathology (infections tumors)
PET (positron emision tomography) ndash the patient is given a radioactive isotope that emits positrons as it decays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of gamma-ray imaging
Bone scan PET image
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
X-ray imaging
Medical diagnosticindustry astronomy
A X-ray tube is a vacuum tube with a cathode and an anode The cathode is heated causing free electrons to be released The electrons flows at high speed to the positively charged anode When the electrons strike a nucleus energy is released in the form of a X-ray radiation The energy (penetreting power) of the X-rays is controlled by a voltage applied across the anode and by a curent applied to the filament in the cathode
The intensity of the X-rays is modified by absorbtion as they pass through the patient and the resulting energy falling develops it much in the same way that light develops photographic film
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Angiography = contrast-enhancement radiography
Angiograms = images of blood vessels
A catheter is inserted into an artery or vein in the groin The catheter is threaded into the blood vessel and guided to the area to be studied When it reaches the area to be studied a X-ray contrast medium is injected through the catheter This enhances contrast of the blood vessels and enables radiologist to see anyirregularities or blockages
X-rays are used in CAT (computerized axial tomography)
X-rays used in industrial processes (examine circuit boards for flows in manifacturing)
Industrial CAT scans are useful when the parts can be penetreted by X-rays
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of X-ray imaging
Chest X-rayAortic angiogram
Head CT Cygnus LoopCircuit boards
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Ultraviolet Band
Litography industrial inspection microscopy biological imaging astronomical observations
Ultraviolet light is used in fluorescence microscopy
Ultraviolet light is not visible to human eye but when a photon of ultraviolet radiation collides with an electron in an atom of a fluorescent material it elevates the electron to a higher energy level After that the electron relaxes to a lower level and emits light in the form of a lower-energy photon in the visible (red) light regionFluorescence = emission of light by a substance that has absorbed light or
other electromagentic radiation of a different wavelengthFlourescence microscope = uses an excitation light to irradiate a prepared specimen
and then it separates the much weaker radiating fluorescent light from the brighter excitation light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Visible and Infrared Bands
Light microscopy astronomy remote sensing industry law enforcement
LANDSAT satellite ndash obtained and transmitted images of the Earth from space for purpose of monitoring environmental conditions on the planet
Weather observations and prediction produce major applications of multispectral image from satellites
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Table 1 ndash Thematic bands in NASArsquos LANDSAT satellite
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Satellite images of Washington DC area in spectral bands of the Table 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Examples of light microscopy
Taxol (anticancer agent)magnified 250X
Cholesterol(40X)
Microprocessor(60X)
Nickel oxidethin film(600X)
Surface of audio CD(1750X)
Organicsuperconductor(450X)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Automated visual inspection of manufactured goods
a bc de f
a ndash a circuit board controllerb ndash packaged pillesc ndash bottlesd ndash air bubbles in clear-plastic producte ndash cerealf ndash image of intraocular implant
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Microwave Band
The dominant aplication of imaging in the microwave band ndash radar
bull radar has the ability to collect data over virtually any region at any time regardless of weather or ambient light conditions
bull some radar waves can penetrate clouds under certain conditions canpenetrate vegetation ice dry sand
bull sometimes radar is the only way to explore inaccessible regions of theEarthrsquos surface
An imaging radar works like a flash camera it provides its own illumination(microwave pulses) to light an area on the ground and take a snapshot image Instead of a camera lens a radar uses an antenna and a digital device torecord the images In a radar image one can see only the microwave energythat was reflected back toward the radar antenna
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Spaceborne radar image of mountains in southeast Tibet
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Imaging in the Radio Band
medicine astronomy
MRI = Magnetic Resonance Imaging
This technique places the patient in a powerful magnet and passes short pulses of radio waves through his or her body
Each pulse causes a responding pulse of radio waves to be emited by the patinet tissues
The location from which these signals originate and their strength are determined by a computer which produces a 2D picture of a section of the patient
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
MRI images of a human knee (left) and spine (right)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Images of the Crab Pulsar covering the electromagnetic spectrum
Gamma X-ray Optical Infrared Radio
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Other Imaging Modalities
acoustic imaging electron microscopy synthetic (computer-generated) imaging
Imaging using sound geological explorations industry medicine
Mineral and oil exploration
For image acquisition over land one of the main approaches is to use a large truck an a large flat steel plate The plate is pressed on the ground by the truck and the truck is vibratedthrough a frequency spectrum up to 100 Hz The strength and the speed of the returning sound waves are determined by the composition of the Earth below the surface These are analysed by a computer and images are generated from the resulting analysis
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - iris
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Biometry - fingerprint
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Face detection and recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Gender identification
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image morphing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Fundamental Steps in DIP
methods whose input and output are images
methods whose inputs are images but whose outputs are attributes extracted from those images
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Outputs are images
bull image acquisition
bull image filtering and enhancement
bull image restoration
bull color image processing
bull wavelets and multiresolution processing
bull compression
bull morphological processing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Outputs are attributes
bull morphological processing
bull segmentation
bull representation and description
bull object recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image acquisition - may involve preprocessing such as scaling
Image enhancement
bull manipulating an image so that the result is more suitable than the original for a specific operation
bull enhancement is problem oriented
bull there is no general sbquotheoryrsquo of image enhancement
bull enhancement use subjective methods for image emprovement
bull enhancement is based on human subjective preferences regarding what is a bdquogoodrdquo enhancement result
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image restoration
bull improving the appearance of an image
bull restoration is objective - the techniques for restoration are based on mathematical or probabilistic models of image degradation
Color image processing
bull fundamental concept in color models
bull basic color processing in a digital domain
Wavelets and multiresolution processing
representing images in various degree of resolution
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Compression
reducing the storage required to save an image or the bandwidth required to transmit it
Morphological processing
bull tools for extracting image components that are useful in the representation and description of shape
bull a transition from processes that output images to processes that outputimage attributes
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Segmentation
bull partitioning an image into its constituents parts or objects
bull autonomous segmentation is one of the most difficult tasks of DIP
bull the more accurate the segmentation the more likley recognition is to succeed
Representation and description (almost always follows segmentation)
bull segmentation produces either the boundary of a region or all the poits in the region itself
bull converting the data produced by segmentation to a form suitable for computer processing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
bull boundary representation the focus is on external shape characteristics such as corners or inflections
bull complete region the focus is on internal properties such as texture or skeletal shape
bull description is also called feature extraction ndash extracting attributes that result in some quantitative information of interest or are basic for differentiating one class of objects from another
Object recognition
the process of assigning a label (eg bdquovehiclerdquo) to an object based on itsdescriptors
Knowledge database
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Simplified diagramof a cross sectionof the human eye
Digital Image ProcessingDigital Image Processing
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Digital Image ProcessingDigital Image Processing
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Three membranes enclose the eye the cornea and sclera outer cover the choroid the retina
The cornea is a tough transparent tissue that covers the anterior surface of the eye
Continuous with the cornea the sclera is an opaque membrane that enclosesthe remainder of the optic globe
The choroid lies directly below the sclera This membrane contains a network of blood vessels (major nutrition of the eye) The choroid is pigmented and help reduce the amount of light entering the eyeThe choroid is divided (at its anterior extreme) into the ciliary body and the irisThe iris contracts and expands to control the amount of light
Digital Image ProcessingDigital Image Processing
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The lens is made up of concentric layers of fibrous cells and is suspended by fibers that attach to ciliary body (60-70 water 6 fat protein) The lens is colored in slightly yellow The lens absorbs approximatively 8 of the visiblelight (infrared and ultraviolet light are absorbed by proteins in lens)
The innermost membrane is the retina When the eye is proper focusedlight from an object outside the eye is imaged on the retinaVision is possible because of the distribution of discrete light receptors on the surface of the retina cones and rods (6-7 milion cones 75-150 milion rods)
Cones located in the central part of the retina (fovea) they are sensitive to colors vision of detail each cone is link to its own nervecone vision = photopic or bright-light vision
Fovea = the place where the image of the object of interest falls on
Digital Image ProcessingDigital Image Processing
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Rods distributed over al the retina surface several rods are contected toa single nerve not specialized in detail vision serve to give a general overall picture of the filed of viewnot involved in color visionsensitive to low level of illumination
Blind spot region without receptors
Digital Image ProcessingDigital Image Processing
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Image formation in the eye
Ordinary photographic camera the lens has fixed focal length focusing atvarious distances is done by modifying the distance between the lens and the image plane (were the film or imaging chip are located)
Human eye the distance between the lens and the retina (the imaging region)is fixed the focal length needed to achieve proper focus is obtained by varyingthe shape of the lens (the fibers in the ciliary body accomplish this flattening or thickening the lens for distant or near objects respectively
distance between lens and retina along visual axix = 17 mm
range of focal length = 14 mm to 17 mm
Digital Image ProcessingDigital Image Processing
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Digital Image ProcessingDigital Image Processing
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Illustration of Mach band effectPercieved intensityis not a simple function of actual intensity
Digital Image ProcessingDigital Image Processing
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All the inner squares have the same intensity but they appear progressively darker as the background becomes lighter
Digital Image ProcessingDigital Image Processing
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Optical illusions
Digital Image ProcessingDigital Image Processing
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ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
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Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
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For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
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the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
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Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
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Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
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Reducing the number of gray levels 256 128 64 32
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Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
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Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
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Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
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binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
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Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
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Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
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The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
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Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
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Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
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Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
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Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
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A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
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Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
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This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
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Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
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a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
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Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
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11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
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1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
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Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
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Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
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Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
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Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
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The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
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Outputs are images
bull image acquisition
bull image filtering and enhancement
bull image restoration
bull color image processing
bull wavelets and multiresolution processing
bull compression
bull morphological processing
Digital Image ProcessingDigital Image Processing
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Outputs are attributes
bull morphological processing
bull segmentation
bull representation and description
bull object recognition
Digital Image ProcessingDigital Image Processing
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Image acquisition - may involve preprocessing such as scaling
Image enhancement
bull manipulating an image so that the result is more suitable than the original for a specific operation
bull enhancement is problem oriented
bull there is no general sbquotheoryrsquo of image enhancement
bull enhancement use subjective methods for image emprovement
bull enhancement is based on human subjective preferences regarding what is a bdquogoodrdquo enhancement result
Digital Image ProcessingDigital Image Processing
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Image restoration
bull improving the appearance of an image
bull restoration is objective - the techniques for restoration are based on mathematical or probabilistic models of image degradation
Color image processing
bull fundamental concept in color models
bull basic color processing in a digital domain
Wavelets and multiresolution processing
representing images in various degree of resolution
Digital Image ProcessingDigital Image Processing
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Compression
reducing the storage required to save an image or the bandwidth required to transmit it
Morphological processing
bull tools for extracting image components that are useful in the representation and description of shape
bull a transition from processes that output images to processes that outputimage attributes
Digital Image ProcessingDigital Image Processing
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Segmentation
bull partitioning an image into its constituents parts or objects
bull autonomous segmentation is one of the most difficult tasks of DIP
bull the more accurate the segmentation the more likley recognition is to succeed
Representation and description (almost always follows segmentation)
bull segmentation produces either the boundary of a region or all the poits in the region itself
bull converting the data produced by segmentation to a form suitable for computer processing
Digital Image ProcessingDigital Image Processing
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bull boundary representation the focus is on external shape characteristics such as corners or inflections
bull complete region the focus is on internal properties such as texture or skeletal shape
bull description is also called feature extraction ndash extracting attributes that result in some quantitative information of interest or are basic for differentiating one class of objects from another
Object recognition
the process of assigning a label (eg bdquovehiclerdquo) to an object based on itsdescriptors
Knowledge database
Digital Image ProcessingDigital Image Processing
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Simplified diagramof a cross sectionof the human eye
Digital Image ProcessingDigital Image Processing
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Digital Image ProcessingDigital Image Processing
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Three membranes enclose the eye the cornea and sclera outer cover the choroid the retina
The cornea is a tough transparent tissue that covers the anterior surface of the eye
Continuous with the cornea the sclera is an opaque membrane that enclosesthe remainder of the optic globe
The choroid lies directly below the sclera This membrane contains a network of blood vessels (major nutrition of the eye) The choroid is pigmented and help reduce the amount of light entering the eyeThe choroid is divided (at its anterior extreme) into the ciliary body and the irisThe iris contracts and expands to control the amount of light
Digital Image ProcessingDigital Image Processing
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The lens is made up of concentric layers of fibrous cells and is suspended by fibers that attach to ciliary body (60-70 water 6 fat protein) The lens is colored in slightly yellow The lens absorbs approximatively 8 of the visiblelight (infrared and ultraviolet light are absorbed by proteins in lens)
The innermost membrane is the retina When the eye is proper focusedlight from an object outside the eye is imaged on the retinaVision is possible because of the distribution of discrete light receptors on the surface of the retina cones and rods (6-7 milion cones 75-150 milion rods)
Cones located in the central part of the retina (fovea) they are sensitive to colors vision of detail each cone is link to its own nervecone vision = photopic or bright-light vision
Fovea = the place where the image of the object of interest falls on
Digital Image ProcessingDigital Image Processing
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Rods distributed over al the retina surface several rods are contected toa single nerve not specialized in detail vision serve to give a general overall picture of the filed of viewnot involved in color visionsensitive to low level of illumination
Blind spot region without receptors
Digital Image ProcessingDigital Image Processing
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Image formation in the eye
Ordinary photographic camera the lens has fixed focal length focusing atvarious distances is done by modifying the distance between the lens and the image plane (were the film or imaging chip are located)
Human eye the distance between the lens and the retina (the imaging region)is fixed the focal length needed to achieve proper focus is obtained by varyingthe shape of the lens (the fibers in the ciliary body accomplish this flattening or thickening the lens for distant or near objects respectively
distance between lens and retina along visual axix = 17 mm
range of focal length = 14 mm to 17 mm
Digital Image ProcessingDigital Image Processing
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Digital Image ProcessingDigital Image Processing
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Illustration of Mach band effectPercieved intensityis not a simple function of actual intensity
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All the inner squares have the same intensity but they appear progressively darker as the background becomes lighter
Digital Image ProcessingDigital Image Processing
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Optical illusions
Digital Image ProcessingDigital Image Processing
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ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
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Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
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For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
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the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
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Digital Image Processing
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Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
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Digital Image Processing
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Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
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Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
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M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
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Digital Image Processing
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Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
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Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
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Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
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Reducing the number of gray levels 256 128 64 32
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Reducing the number of gray levels 16 8 4 2
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Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
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Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
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Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
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Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
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Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
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Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
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binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
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A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
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Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
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Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
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The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
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8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
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Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
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Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
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1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
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Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
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A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
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(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Outputs are attributes
bull morphological processing
bull segmentation
bull representation and description
bull object recognition
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image acquisition - may involve preprocessing such as scaling
Image enhancement
bull manipulating an image so that the result is more suitable than the original for a specific operation
bull enhancement is problem oriented
bull there is no general sbquotheoryrsquo of image enhancement
bull enhancement use subjective methods for image emprovement
bull enhancement is based on human subjective preferences regarding what is a bdquogoodrdquo enhancement result
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image restoration
bull improving the appearance of an image
bull restoration is objective - the techniques for restoration are based on mathematical or probabilistic models of image degradation
Color image processing
bull fundamental concept in color models
bull basic color processing in a digital domain
Wavelets and multiresolution processing
representing images in various degree of resolution
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Compression
reducing the storage required to save an image or the bandwidth required to transmit it
Morphological processing
bull tools for extracting image components that are useful in the representation and description of shape
bull a transition from processes that output images to processes that outputimage attributes
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Segmentation
bull partitioning an image into its constituents parts or objects
bull autonomous segmentation is one of the most difficult tasks of DIP
bull the more accurate the segmentation the more likley recognition is to succeed
Representation and description (almost always follows segmentation)
bull segmentation produces either the boundary of a region or all the poits in the region itself
bull converting the data produced by segmentation to a form suitable for computer processing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
bull boundary representation the focus is on external shape characteristics such as corners or inflections
bull complete region the focus is on internal properties such as texture or skeletal shape
bull description is also called feature extraction ndash extracting attributes that result in some quantitative information of interest or are basic for differentiating one class of objects from another
Object recognition
the process of assigning a label (eg bdquovehiclerdquo) to an object based on itsdescriptors
Knowledge database
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Simplified diagramof a cross sectionof the human eye
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Three membranes enclose the eye the cornea and sclera outer cover the choroid the retina
The cornea is a tough transparent tissue that covers the anterior surface of the eye
Continuous with the cornea the sclera is an opaque membrane that enclosesthe remainder of the optic globe
The choroid lies directly below the sclera This membrane contains a network of blood vessels (major nutrition of the eye) The choroid is pigmented and help reduce the amount of light entering the eyeThe choroid is divided (at its anterior extreme) into the ciliary body and the irisThe iris contracts and expands to control the amount of light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The lens is made up of concentric layers of fibrous cells and is suspended by fibers that attach to ciliary body (60-70 water 6 fat protein) The lens is colored in slightly yellow The lens absorbs approximatively 8 of the visiblelight (infrared and ultraviolet light are absorbed by proteins in lens)
The innermost membrane is the retina When the eye is proper focusedlight from an object outside the eye is imaged on the retinaVision is possible because of the distribution of discrete light receptors on the surface of the retina cones and rods (6-7 milion cones 75-150 milion rods)
Cones located in the central part of the retina (fovea) they are sensitive to colors vision of detail each cone is link to its own nervecone vision = photopic or bright-light vision
Fovea = the place where the image of the object of interest falls on
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Rods distributed over al the retina surface several rods are contected toa single nerve not specialized in detail vision serve to give a general overall picture of the filed of viewnot involved in color visionsensitive to low level of illumination
Blind spot region without receptors
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image formation in the eye
Ordinary photographic camera the lens has fixed focal length focusing atvarious distances is done by modifying the distance between the lens and the image plane (were the film or imaging chip are located)
Human eye the distance between the lens and the retina (the imaging region)is fixed the focal length needed to achieve proper focus is obtained by varyingthe shape of the lens (the fibers in the ciliary body accomplish this flattening or thickening the lens for distant or near objects respectively
distance between lens and retina along visual axix = 17 mm
range of focal length = 14 mm to 17 mm
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Illustration of Mach band effectPercieved intensityis not a simple function of actual intensity
Digital Image ProcessingDigital Image Processing
Week 1Week 1
All the inner squares have the same intensity but they appear progressively darker as the background becomes lighter
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Optical illusions
Digital Image ProcessingDigital Image Processing
Week 1Week 1
ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
Week 1Week 1
For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image acquisition - may involve preprocessing such as scaling
Image enhancement
bull manipulating an image so that the result is more suitable than the original for a specific operation
bull enhancement is problem oriented
bull there is no general sbquotheoryrsquo of image enhancement
bull enhancement use subjective methods for image emprovement
bull enhancement is based on human subjective preferences regarding what is a bdquogoodrdquo enhancement result
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image restoration
bull improving the appearance of an image
bull restoration is objective - the techniques for restoration are based on mathematical or probabilistic models of image degradation
Color image processing
bull fundamental concept in color models
bull basic color processing in a digital domain
Wavelets and multiresolution processing
representing images in various degree of resolution
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Compression
reducing the storage required to save an image or the bandwidth required to transmit it
Morphological processing
bull tools for extracting image components that are useful in the representation and description of shape
bull a transition from processes that output images to processes that outputimage attributes
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Segmentation
bull partitioning an image into its constituents parts or objects
bull autonomous segmentation is one of the most difficult tasks of DIP
bull the more accurate the segmentation the more likley recognition is to succeed
Representation and description (almost always follows segmentation)
bull segmentation produces either the boundary of a region or all the poits in the region itself
bull converting the data produced by segmentation to a form suitable for computer processing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
bull boundary representation the focus is on external shape characteristics such as corners or inflections
bull complete region the focus is on internal properties such as texture or skeletal shape
bull description is also called feature extraction ndash extracting attributes that result in some quantitative information of interest or are basic for differentiating one class of objects from another
Object recognition
the process of assigning a label (eg bdquovehiclerdquo) to an object based on itsdescriptors
Knowledge database
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Simplified diagramof a cross sectionof the human eye
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Three membranes enclose the eye the cornea and sclera outer cover the choroid the retina
The cornea is a tough transparent tissue that covers the anterior surface of the eye
Continuous with the cornea the sclera is an opaque membrane that enclosesthe remainder of the optic globe
The choroid lies directly below the sclera This membrane contains a network of blood vessels (major nutrition of the eye) The choroid is pigmented and help reduce the amount of light entering the eyeThe choroid is divided (at its anterior extreme) into the ciliary body and the irisThe iris contracts and expands to control the amount of light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The lens is made up of concentric layers of fibrous cells and is suspended by fibers that attach to ciliary body (60-70 water 6 fat protein) The lens is colored in slightly yellow The lens absorbs approximatively 8 of the visiblelight (infrared and ultraviolet light are absorbed by proteins in lens)
The innermost membrane is the retina When the eye is proper focusedlight from an object outside the eye is imaged on the retinaVision is possible because of the distribution of discrete light receptors on the surface of the retina cones and rods (6-7 milion cones 75-150 milion rods)
Cones located in the central part of the retina (fovea) they are sensitive to colors vision of detail each cone is link to its own nervecone vision = photopic or bright-light vision
Fovea = the place where the image of the object of interest falls on
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Rods distributed over al the retina surface several rods are contected toa single nerve not specialized in detail vision serve to give a general overall picture of the filed of viewnot involved in color visionsensitive to low level of illumination
Blind spot region without receptors
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image formation in the eye
Ordinary photographic camera the lens has fixed focal length focusing atvarious distances is done by modifying the distance between the lens and the image plane (were the film or imaging chip are located)
Human eye the distance between the lens and the retina (the imaging region)is fixed the focal length needed to achieve proper focus is obtained by varyingthe shape of the lens (the fibers in the ciliary body accomplish this flattening or thickening the lens for distant or near objects respectively
distance between lens and retina along visual axix = 17 mm
range of focal length = 14 mm to 17 mm
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Illustration of Mach band effectPercieved intensityis not a simple function of actual intensity
Digital Image ProcessingDigital Image Processing
Week 1Week 1
All the inner squares have the same intensity but they appear progressively darker as the background becomes lighter
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Optical illusions
Digital Image ProcessingDigital Image Processing
Week 1Week 1
ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
Week 1Week 1
For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image restoration
bull improving the appearance of an image
bull restoration is objective - the techniques for restoration are based on mathematical or probabilistic models of image degradation
Color image processing
bull fundamental concept in color models
bull basic color processing in a digital domain
Wavelets and multiresolution processing
representing images in various degree of resolution
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Compression
reducing the storage required to save an image or the bandwidth required to transmit it
Morphological processing
bull tools for extracting image components that are useful in the representation and description of shape
bull a transition from processes that output images to processes that outputimage attributes
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Segmentation
bull partitioning an image into its constituents parts or objects
bull autonomous segmentation is one of the most difficult tasks of DIP
bull the more accurate the segmentation the more likley recognition is to succeed
Representation and description (almost always follows segmentation)
bull segmentation produces either the boundary of a region or all the poits in the region itself
bull converting the data produced by segmentation to a form suitable for computer processing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
bull boundary representation the focus is on external shape characteristics such as corners or inflections
bull complete region the focus is on internal properties such as texture or skeletal shape
bull description is also called feature extraction ndash extracting attributes that result in some quantitative information of interest or are basic for differentiating one class of objects from another
Object recognition
the process of assigning a label (eg bdquovehiclerdquo) to an object based on itsdescriptors
Knowledge database
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Simplified diagramof a cross sectionof the human eye
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Three membranes enclose the eye the cornea and sclera outer cover the choroid the retina
The cornea is a tough transparent tissue that covers the anterior surface of the eye
Continuous with the cornea the sclera is an opaque membrane that enclosesthe remainder of the optic globe
The choroid lies directly below the sclera This membrane contains a network of blood vessels (major nutrition of the eye) The choroid is pigmented and help reduce the amount of light entering the eyeThe choroid is divided (at its anterior extreme) into the ciliary body and the irisThe iris contracts and expands to control the amount of light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The lens is made up of concentric layers of fibrous cells and is suspended by fibers that attach to ciliary body (60-70 water 6 fat protein) The lens is colored in slightly yellow The lens absorbs approximatively 8 of the visiblelight (infrared and ultraviolet light are absorbed by proteins in lens)
The innermost membrane is the retina When the eye is proper focusedlight from an object outside the eye is imaged on the retinaVision is possible because of the distribution of discrete light receptors on the surface of the retina cones and rods (6-7 milion cones 75-150 milion rods)
Cones located in the central part of the retina (fovea) they are sensitive to colors vision of detail each cone is link to its own nervecone vision = photopic or bright-light vision
Fovea = the place where the image of the object of interest falls on
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Rods distributed over al the retina surface several rods are contected toa single nerve not specialized in detail vision serve to give a general overall picture of the filed of viewnot involved in color visionsensitive to low level of illumination
Blind spot region without receptors
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image formation in the eye
Ordinary photographic camera the lens has fixed focal length focusing atvarious distances is done by modifying the distance between the lens and the image plane (were the film or imaging chip are located)
Human eye the distance between the lens and the retina (the imaging region)is fixed the focal length needed to achieve proper focus is obtained by varyingthe shape of the lens (the fibers in the ciliary body accomplish this flattening or thickening the lens for distant or near objects respectively
distance between lens and retina along visual axix = 17 mm
range of focal length = 14 mm to 17 mm
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Illustration of Mach band effectPercieved intensityis not a simple function of actual intensity
Digital Image ProcessingDigital Image Processing
Week 1Week 1
All the inner squares have the same intensity but they appear progressively darker as the background becomes lighter
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Optical illusions
Digital Image ProcessingDigital Image Processing
Week 1Week 1
ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
Week 1Week 1
For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Compression
reducing the storage required to save an image or the bandwidth required to transmit it
Morphological processing
bull tools for extracting image components that are useful in the representation and description of shape
bull a transition from processes that output images to processes that outputimage attributes
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Segmentation
bull partitioning an image into its constituents parts or objects
bull autonomous segmentation is one of the most difficult tasks of DIP
bull the more accurate the segmentation the more likley recognition is to succeed
Representation and description (almost always follows segmentation)
bull segmentation produces either the boundary of a region or all the poits in the region itself
bull converting the data produced by segmentation to a form suitable for computer processing
Digital Image ProcessingDigital Image Processing
Week 1Week 1
bull boundary representation the focus is on external shape characteristics such as corners or inflections
bull complete region the focus is on internal properties such as texture or skeletal shape
bull description is also called feature extraction ndash extracting attributes that result in some quantitative information of interest or are basic for differentiating one class of objects from another
Object recognition
the process of assigning a label (eg bdquovehiclerdquo) to an object based on itsdescriptors
Knowledge database
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Simplified diagramof a cross sectionof the human eye
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Three membranes enclose the eye the cornea and sclera outer cover the choroid the retina
The cornea is a tough transparent tissue that covers the anterior surface of the eye
Continuous with the cornea the sclera is an opaque membrane that enclosesthe remainder of the optic globe
The choroid lies directly below the sclera This membrane contains a network of blood vessels (major nutrition of the eye) The choroid is pigmented and help reduce the amount of light entering the eyeThe choroid is divided (at its anterior extreme) into the ciliary body and the irisThe iris contracts and expands to control the amount of light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The lens is made up of concentric layers of fibrous cells and is suspended by fibers that attach to ciliary body (60-70 water 6 fat protein) The lens is colored in slightly yellow The lens absorbs approximatively 8 of the visiblelight (infrared and ultraviolet light are absorbed by proteins in lens)
The innermost membrane is the retina When the eye is proper focusedlight from an object outside the eye is imaged on the retinaVision is possible because of the distribution of discrete light receptors on the surface of the retina cones and rods (6-7 milion cones 75-150 milion rods)
Cones located in the central part of the retina (fovea) they are sensitive to colors vision of detail each cone is link to its own nervecone vision = photopic or bright-light vision
Fovea = the place where the image of the object of interest falls on
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Rods distributed over al the retina surface several rods are contected toa single nerve not specialized in detail vision serve to give a general overall picture of the filed of viewnot involved in color visionsensitive to low level of illumination
Blind spot region without receptors
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image formation in the eye
Ordinary photographic camera the lens has fixed focal length focusing atvarious distances is done by modifying the distance between the lens and the image plane (were the film or imaging chip are located)
Human eye the distance between the lens and the retina (the imaging region)is fixed the focal length needed to achieve proper focus is obtained by varyingthe shape of the lens (the fibers in the ciliary body accomplish this flattening or thickening the lens for distant or near objects respectively
distance between lens and retina along visual axix = 17 mm
range of focal length = 14 mm to 17 mm
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Illustration of Mach band effectPercieved intensityis not a simple function of actual intensity
Digital Image ProcessingDigital Image Processing
Week 1Week 1
All the inner squares have the same intensity but they appear progressively darker as the background becomes lighter
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Optical illusions
Digital Image ProcessingDigital Image Processing
Week 1Week 1
ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
Week 1Week 1
For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
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Segmentation
bull partitioning an image into its constituents parts or objects
bull autonomous segmentation is one of the most difficult tasks of DIP
bull the more accurate the segmentation the more likley recognition is to succeed
Representation and description (almost always follows segmentation)
bull segmentation produces either the boundary of a region or all the poits in the region itself
bull converting the data produced by segmentation to a form suitable for computer processing
Digital Image ProcessingDigital Image Processing
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bull boundary representation the focus is on external shape characteristics such as corners or inflections
bull complete region the focus is on internal properties such as texture or skeletal shape
bull description is also called feature extraction ndash extracting attributes that result in some quantitative information of interest or are basic for differentiating one class of objects from another
Object recognition
the process of assigning a label (eg bdquovehiclerdquo) to an object based on itsdescriptors
Knowledge database
Digital Image ProcessingDigital Image Processing
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Simplified diagramof a cross sectionof the human eye
Digital Image ProcessingDigital Image Processing
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Digital Image ProcessingDigital Image Processing
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Three membranes enclose the eye the cornea and sclera outer cover the choroid the retina
The cornea is a tough transparent tissue that covers the anterior surface of the eye
Continuous with the cornea the sclera is an opaque membrane that enclosesthe remainder of the optic globe
The choroid lies directly below the sclera This membrane contains a network of blood vessels (major nutrition of the eye) The choroid is pigmented and help reduce the amount of light entering the eyeThe choroid is divided (at its anterior extreme) into the ciliary body and the irisThe iris contracts and expands to control the amount of light
Digital Image ProcessingDigital Image Processing
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The lens is made up of concentric layers of fibrous cells and is suspended by fibers that attach to ciliary body (60-70 water 6 fat protein) The lens is colored in slightly yellow The lens absorbs approximatively 8 of the visiblelight (infrared and ultraviolet light are absorbed by proteins in lens)
The innermost membrane is the retina When the eye is proper focusedlight from an object outside the eye is imaged on the retinaVision is possible because of the distribution of discrete light receptors on the surface of the retina cones and rods (6-7 milion cones 75-150 milion rods)
Cones located in the central part of the retina (fovea) they are sensitive to colors vision of detail each cone is link to its own nervecone vision = photopic or bright-light vision
Fovea = the place where the image of the object of interest falls on
Digital Image ProcessingDigital Image Processing
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Rods distributed over al the retina surface several rods are contected toa single nerve not specialized in detail vision serve to give a general overall picture of the filed of viewnot involved in color visionsensitive to low level of illumination
Blind spot region without receptors
Digital Image ProcessingDigital Image Processing
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Image formation in the eye
Ordinary photographic camera the lens has fixed focal length focusing atvarious distances is done by modifying the distance between the lens and the image plane (were the film or imaging chip are located)
Human eye the distance between the lens and the retina (the imaging region)is fixed the focal length needed to achieve proper focus is obtained by varyingthe shape of the lens (the fibers in the ciliary body accomplish this flattening or thickening the lens for distant or near objects respectively
distance between lens and retina along visual axix = 17 mm
range of focal length = 14 mm to 17 mm
Digital Image ProcessingDigital Image Processing
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Digital Image ProcessingDigital Image Processing
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Illustration of Mach band effectPercieved intensityis not a simple function of actual intensity
Digital Image ProcessingDigital Image Processing
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All the inner squares have the same intensity but they appear progressively darker as the background becomes lighter
Digital Image ProcessingDigital Image Processing
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Optical illusions
Digital Image ProcessingDigital Image Processing
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ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
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For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
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the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
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Digital Image Processing
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Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
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Reducing the number of gray levels 256 128 64 32
Digital Image Processing
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Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
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A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
bull boundary representation the focus is on external shape characteristics such as corners or inflections
bull complete region the focus is on internal properties such as texture or skeletal shape
bull description is also called feature extraction ndash extracting attributes that result in some quantitative information of interest or are basic for differentiating one class of objects from another
Object recognition
the process of assigning a label (eg bdquovehiclerdquo) to an object based on itsdescriptors
Knowledge database
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Simplified diagramof a cross sectionof the human eye
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Three membranes enclose the eye the cornea and sclera outer cover the choroid the retina
The cornea is a tough transparent tissue that covers the anterior surface of the eye
Continuous with the cornea the sclera is an opaque membrane that enclosesthe remainder of the optic globe
The choroid lies directly below the sclera This membrane contains a network of blood vessels (major nutrition of the eye) The choroid is pigmented and help reduce the amount of light entering the eyeThe choroid is divided (at its anterior extreme) into the ciliary body and the irisThe iris contracts and expands to control the amount of light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The lens is made up of concentric layers of fibrous cells and is suspended by fibers that attach to ciliary body (60-70 water 6 fat protein) The lens is colored in slightly yellow The lens absorbs approximatively 8 of the visiblelight (infrared and ultraviolet light are absorbed by proteins in lens)
The innermost membrane is the retina When the eye is proper focusedlight from an object outside the eye is imaged on the retinaVision is possible because of the distribution of discrete light receptors on the surface of the retina cones and rods (6-7 milion cones 75-150 milion rods)
Cones located in the central part of the retina (fovea) they are sensitive to colors vision of detail each cone is link to its own nervecone vision = photopic or bright-light vision
Fovea = the place where the image of the object of interest falls on
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Rods distributed over al the retina surface several rods are contected toa single nerve not specialized in detail vision serve to give a general overall picture of the filed of viewnot involved in color visionsensitive to low level of illumination
Blind spot region without receptors
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image formation in the eye
Ordinary photographic camera the lens has fixed focal length focusing atvarious distances is done by modifying the distance between the lens and the image plane (were the film or imaging chip are located)
Human eye the distance between the lens and the retina (the imaging region)is fixed the focal length needed to achieve proper focus is obtained by varyingthe shape of the lens (the fibers in the ciliary body accomplish this flattening or thickening the lens for distant or near objects respectively
distance between lens and retina along visual axix = 17 mm
range of focal length = 14 mm to 17 mm
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Illustration of Mach band effectPercieved intensityis not a simple function of actual intensity
Digital Image ProcessingDigital Image Processing
Week 1Week 1
All the inner squares have the same intensity but they appear progressively darker as the background becomes lighter
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Optical illusions
Digital Image ProcessingDigital Image Processing
Week 1Week 1
ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
Week 1Week 1
For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Simplified diagramof a cross sectionof the human eye
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Three membranes enclose the eye the cornea and sclera outer cover the choroid the retina
The cornea is a tough transparent tissue that covers the anterior surface of the eye
Continuous with the cornea the sclera is an opaque membrane that enclosesthe remainder of the optic globe
The choroid lies directly below the sclera This membrane contains a network of blood vessels (major nutrition of the eye) The choroid is pigmented and help reduce the amount of light entering the eyeThe choroid is divided (at its anterior extreme) into the ciliary body and the irisThe iris contracts and expands to control the amount of light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The lens is made up of concentric layers of fibrous cells and is suspended by fibers that attach to ciliary body (60-70 water 6 fat protein) The lens is colored in slightly yellow The lens absorbs approximatively 8 of the visiblelight (infrared and ultraviolet light are absorbed by proteins in lens)
The innermost membrane is the retina When the eye is proper focusedlight from an object outside the eye is imaged on the retinaVision is possible because of the distribution of discrete light receptors on the surface of the retina cones and rods (6-7 milion cones 75-150 milion rods)
Cones located in the central part of the retina (fovea) they are sensitive to colors vision of detail each cone is link to its own nervecone vision = photopic or bright-light vision
Fovea = the place where the image of the object of interest falls on
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Rods distributed over al the retina surface several rods are contected toa single nerve not specialized in detail vision serve to give a general overall picture of the filed of viewnot involved in color visionsensitive to low level of illumination
Blind spot region without receptors
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image formation in the eye
Ordinary photographic camera the lens has fixed focal length focusing atvarious distances is done by modifying the distance between the lens and the image plane (were the film or imaging chip are located)
Human eye the distance between the lens and the retina (the imaging region)is fixed the focal length needed to achieve proper focus is obtained by varyingthe shape of the lens (the fibers in the ciliary body accomplish this flattening or thickening the lens for distant or near objects respectively
distance between lens and retina along visual axix = 17 mm
range of focal length = 14 mm to 17 mm
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Illustration of Mach band effectPercieved intensityis not a simple function of actual intensity
Digital Image ProcessingDigital Image Processing
Week 1Week 1
All the inner squares have the same intensity but they appear progressively darker as the background becomes lighter
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Optical illusions
Digital Image ProcessingDigital Image Processing
Week 1Week 1
ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
Week 1Week 1
For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Three membranes enclose the eye the cornea and sclera outer cover the choroid the retina
The cornea is a tough transparent tissue that covers the anterior surface of the eye
Continuous with the cornea the sclera is an opaque membrane that enclosesthe remainder of the optic globe
The choroid lies directly below the sclera This membrane contains a network of blood vessels (major nutrition of the eye) The choroid is pigmented and help reduce the amount of light entering the eyeThe choroid is divided (at its anterior extreme) into the ciliary body and the irisThe iris contracts and expands to control the amount of light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The lens is made up of concentric layers of fibrous cells and is suspended by fibers that attach to ciliary body (60-70 water 6 fat protein) The lens is colored in slightly yellow The lens absorbs approximatively 8 of the visiblelight (infrared and ultraviolet light are absorbed by proteins in lens)
The innermost membrane is the retina When the eye is proper focusedlight from an object outside the eye is imaged on the retinaVision is possible because of the distribution of discrete light receptors on the surface of the retina cones and rods (6-7 milion cones 75-150 milion rods)
Cones located in the central part of the retina (fovea) they are sensitive to colors vision of detail each cone is link to its own nervecone vision = photopic or bright-light vision
Fovea = the place where the image of the object of interest falls on
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Rods distributed over al the retina surface several rods are contected toa single nerve not specialized in detail vision serve to give a general overall picture of the filed of viewnot involved in color visionsensitive to low level of illumination
Blind spot region without receptors
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image formation in the eye
Ordinary photographic camera the lens has fixed focal length focusing atvarious distances is done by modifying the distance between the lens and the image plane (were the film or imaging chip are located)
Human eye the distance between the lens and the retina (the imaging region)is fixed the focal length needed to achieve proper focus is obtained by varyingthe shape of the lens (the fibers in the ciliary body accomplish this flattening or thickening the lens for distant or near objects respectively
distance between lens and retina along visual axix = 17 mm
range of focal length = 14 mm to 17 mm
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Illustration of Mach band effectPercieved intensityis not a simple function of actual intensity
Digital Image ProcessingDigital Image Processing
Week 1Week 1
All the inner squares have the same intensity but they appear progressively darker as the background becomes lighter
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Optical illusions
Digital Image ProcessingDigital Image Processing
Week 1Week 1
ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
Week 1Week 1
For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Three membranes enclose the eye the cornea and sclera outer cover the choroid the retina
The cornea is a tough transparent tissue that covers the anterior surface of the eye
Continuous with the cornea the sclera is an opaque membrane that enclosesthe remainder of the optic globe
The choroid lies directly below the sclera This membrane contains a network of blood vessels (major nutrition of the eye) The choroid is pigmented and help reduce the amount of light entering the eyeThe choroid is divided (at its anterior extreme) into the ciliary body and the irisThe iris contracts and expands to control the amount of light
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The lens is made up of concentric layers of fibrous cells and is suspended by fibers that attach to ciliary body (60-70 water 6 fat protein) The lens is colored in slightly yellow The lens absorbs approximatively 8 of the visiblelight (infrared and ultraviolet light are absorbed by proteins in lens)
The innermost membrane is the retina When the eye is proper focusedlight from an object outside the eye is imaged on the retinaVision is possible because of the distribution of discrete light receptors on the surface of the retina cones and rods (6-7 milion cones 75-150 milion rods)
Cones located in the central part of the retina (fovea) they are sensitive to colors vision of detail each cone is link to its own nervecone vision = photopic or bright-light vision
Fovea = the place where the image of the object of interest falls on
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Rods distributed over al the retina surface several rods are contected toa single nerve not specialized in detail vision serve to give a general overall picture of the filed of viewnot involved in color visionsensitive to low level of illumination
Blind spot region without receptors
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image formation in the eye
Ordinary photographic camera the lens has fixed focal length focusing atvarious distances is done by modifying the distance between the lens and the image plane (were the film or imaging chip are located)
Human eye the distance between the lens and the retina (the imaging region)is fixed the focal length needed to achieve proper focus is obtained by varyingthe shape of the lens (the fibers in the ciliary body accomplish this flattening or thickening the lens for distant or near objects respectively
distance between lens and retina along visual axix = 17 mm
range of focal length = 14 mm to 17 mm
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Illustration of Mach band effectPercieved intensityis not a simple function of actual intensity
Digital Image ProcessingDigital Image Processing
Week 1Week 1
All the inner squares have the same intensity but they appear progressively darker as the background becomes lighter
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Optical illusions
Digital Image ProcessingDigital Image Processing
Week 1Week 1
ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
Week 1Week 1
For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
The lens is made up of concentric layers of fibrous cells and is suspended by fibers that attach to ciliary body (60-70 water 6 fat protein) The lens is colored in slightly yellow The lens absorbs approximatively 8 of the visiblelight (infrared and ultraviolet light are absorbed by proteins in lens)
The innermost membrane is the retina When the eye is proper focusedlight from an object outside the eye is imaged on the retinaVision is possible because of the distribution of discrete light receptors on the surface of the retina cones and rods (6-7 milion cones 75-150 milion rods)
Cones located in the central part of the retina (fovea) they are sensitive to colors vision of detail each cone is link to its own nervecone vision = photopic or bright-light vision
Fovea = the place where the image of the object of interest falls on
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Rods distributed over al the retina surface several rods are contected toa single nerve not specialized in detail vision serve to give a general overall picture of the filed of viewnot involved in color visionsensitive to low level of illumination
Blind spot region without receptors
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image formation in the eye
Ordinary photographic camera the lens has fixed focal length focusing atvarious distances is done by modifying the distance between the lens and the image plane (were the film or imaging chip are located)
Human eye the distance between the lens and the retina (the imaging region)is fixed the focal length needed to achieve proper focus is obtained by varyingthe shape of the lens (the fibers in the ciliary body accomplish this flattening or thickening the lens for distant or near objects respectively
distance between lens and retina along visual axix = 17 mm
range of focal length = 14 mm to 17 mm
Digital Image ProcessingDigital Image Processing
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Digital Image ProcessingDigital Image Processing
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Illustration of Mach band effectPercieved intensityis not a simple function of actual intensity
Digital Image ProcessingDigital Image Processing
Week 1Week 1
All the inner squares have the same intensity but they appear progressively darker as the background becomes lighter
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Optical illusions
Digital Image ProcessingDigital Image Processing
Week 1Week 1
ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
Week 1Week 1
For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Rods distributed over al the retina surface several rods are contected toa single nerve not specialized in detail vision serve to give a general overall picture of the filed of viewnot involved in color visionsensitive to low level of illumination
Blind spot region without receptors
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image formation in the eye
Ordinary photographic camera the lens has fixed focal length focusing atvarious distances is done by modifying the distance between the lens and the image plane (were the film or imaging chip are located)
Human eye the distance between the lens and the retina (the imaging region)is fixed the focal length needed to achieve proper focus is obtained by varyingthe shape of the lens (the fibers in the ciliary body accomplish this flattening or thickening the lens for distant or near objects respectively
distance between lens and retina along visual axix = 17 mm
range of focal length = 14 mm to 17 mm
Digital Image ProcessingDigital Image Processing
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Digital Image ProcessingDigital Image Processing
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Illustration of Mach band effectPercieved intensityis not a simple function of actual intensity
Digital Image ProcessingDigital Image Processing
Week 1Week 1
All the inner squares have the same intensity but they appear progressively darker as the background becomes lighter
Digital Image ProcessingDigital Image Processing
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Optical illusions
Digital Image ProcessingDigital Image Processing
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ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
Week 1Week 1
For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Image formation in the eye
Ordinary photographic camera the lens has fixed focal length focusing atvarious distances is done by modifying the distance between the lens and the image plane (were the film or imaging chip are located)
Human eye the distance between the lens and the retina (the imaging region)is fixed the focal length needed to achieve proper focus is obtained by varyingthe shape of the lens (the fibers in the ciliary body accomplish this flattening or thickening the lens for distant or near objects respectively
distance between lens and retina along visual axix = 17 mm
range of focal length = 14 mm to 17 mm
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Illustration of Mach band effectPercieved intensityis not a simple function of actual intensity
Digital Image ProcessingDigital Image Processing
Week 1Week 1
All the inner squares have the same intensity but they appear progressively darker as the background becomes lighter
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Optical illusions
Digital Image ProcessingDigital Image Processing
Week 1Week 1
ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
Week 1Week 1
For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Illustration of Mach band effectPercieved intensityis not a simple function of actual intensity
Digital Image ProcessingDigital Image Processing
Week 1Week 1
All the inner squares have the same intensity but they appear progressively darker as the background becomes lighter
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Optical illusions
Digital Image ProcessingDigital Image Processing
Week 1Week 1
ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
Week 1Week 1
For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Illustration of Mach band effectPercieved intensityis not a simple function of actual intensity
Digital Image ProcessingDigital Image Processing
Week 1Week 1
All the inner squares have the same intensity but they appear progressively darker as the background becomes lighter
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Optical illusions
Digital Image ProcessingDigital Image Processing
Week 1Week 1
ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
Week 1Week 1
For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
All the inner squares have the same intensity but they appear progressively darker as the background becomes lighter
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Optical illusions
Digital Image ProcessingDigital Image Processing
Week 1Week 1
ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
Week 1Week 1
For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Optical illusions
Digital Image ProcessingDigital Image Processing
Week 1Week 1
ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
Week 1Week 1
For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
ALBASTRU BLUEVERDE GREENGALBEN YELLOW ROSU REDPORTOCALIU ORANGEGALBEN YELLOWVISINIU BURGUNDYALB WHITE
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
Week 1Week 1
For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Lightbullachromatic or monochromatic light - light that is void of color the attribute of such light is its intensity or amount
gray level is used to describe monochromatic intensitybecause it ranges from black to grays and to white bull chromatic light spans the electromagnetic energy spectrumfrom approximately 043 to 079 μm
quantities that describe the quality of a chromatic light source radiance
the total amount of energy that flows from the light source and it isusually measured in watts (W) luminance
measured in lumens (lm) gives a measure of the amount of energy an observer perceives from a light source
Digital Image ProcessingDigital Image Processing
Week 1Week 1
For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
For example light emitted from a source operating in the far infraredregion of the spectrum could have significant energy (radiance) butan observer would hardly perceive it its luminance would be almostzero
brightnessa subjective descriptor of light perception that is practicallyimpossible to measure It embodies the achromatic notion ofintensity and is one of the key factors in describing color sensation
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
the physical meaning is determined by the source of the image
( )f D f x y
Image generated from a physical process f(xy) ndash proportional to the energy radiated by the physical source
f(xy) ndash characterized by two components
i(xy) = illumination component the amount of source illumination incident on the scene being viewed
r(xy) = reflectance component the amount of illumination reflected by the objects in the scene
( ) ( ) ( )
0 ( ) 0 ( ) 1
f x y i x y r x y
i x y r x y
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
r(xy)=0 - total absorption r(xy)=1 - total reflectance
i(xy) ndash determined by the illumination source
r(xy) ndash determined by the characteristics of the imaged objects
is called gray (or intensity) scale
In practice
min 0 0 max min min min max max max( ) L l f x y L L i r L i r
indoor values without additional illuminationmin max10 1000L L
black whitemin max0 1 0 1 0 1L L L L l l L
min maxL L
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image ProcessingDigital Image Processing
Week 1Week 1
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Image Sampling and Quantization
- the output of the sensors is a continuous voltage waveform related to the sensed
scene
converting a continuous image f to digital form
- digitizing (x y) is called sampling
- digitizing f(x y) is called quantization
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
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[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
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The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Continuous image projected onto a sensor array Result of image sampling and quantization
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Representing Digital Images (xy) x = 01hellipM-1 y = 01hellipN-1 ndash spatial variables or spatial coordinates
(00) (01) (0 1)(10) (11) (1 1)
( )
( 10) ( 11) ( 1 1)
f f f Nf f f N
f x y
f M f M f M N
image element pixel
00 01 0 1
10 11 1 1
10 11 1 1
( ) ( )
N
i jN M N
i j
M M M N
a a aa f x i y j f i ja a a
Aa
a a a
f(00) ndash the upper left corner of the image
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
M N ge 0 L=2k
[0 1]i j i ja a L
Dynamic range of an image = the ratio of the maximum measurable intensity to the minimum detectable intensity level in the system Upper limit ndash determined by saturation lower limit - noise
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Number of bits required to store a digitized image
for 2 b M N k M N b N k
When an image can have 2k intensity levels the image is referred as a k-bit image 256 discrete intensity values ndash 8-bit image
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Spatial and Intensity Resolution Spatial resolution ndash the smallest discernible detail in an image
Measures line pairs per unit distance dots (pixels) per unit distance
Image resolution = the largest number of discernible line pairs per unit distance
(eg 100 line pairs per mm)
Dots per unit distance are commonly used in printing and publishing
In US the measure is expressed in dots per inch (dpi)
(newspapers are printed with 75 dpi glossy brochures at 175 dpi)
Intensity resolution ndash the smallest discernible change in intensity level
The number of intensity levels (L) is determined by hardware considerations
L=2k ndash most common k = 8
Intensity resolution in practice is given by k (number of bits used to quantize intensity)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Fig1 Reducing spatial resolution 1250 dpi(upper left) 300 dpi (upper right)
150 dpi (lower left) 72 dpi (lower right)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Reducing the number of gray levels 256 128 64 32
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
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Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
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A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Reducing the number of gray levels 16 8 4 2
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Image Interpolation - used in zooming shrinking rotating and geometric corrections
Shrinking zooming ndash image resizing ndash image resampling methods
Interpolation is the process of using known data to estimate values at unknown locations
Suppose we have an image of size 500 times 500 pixels that has to be enlarged 15 times to
750times750 pixels One way to do this is to create an imaginary 750 times 750 grid with the
same spacing as the original and then shrink it so that it fits exactly over the original
image The pixel spacing in the 750 times 750 grid will be less than in the original image
Problem assignment of intensity-level in the new 750 times 750 grid
Nearest neighbor interpolation assign for every point in the new grid (750 times 750) the
intensity of the closest pixel (nearest neighbor) from the oldoriginal grid (500 times 500)
This technique has the tendency to produce undesirable effects like severe distortion of
straight edges
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Bilinear interpolation ndash assign for the new (x y) location the following intensity ( )v x y a x b y c x y d
where the four coefficients are determined from the 4 equations in 4 unknowns that can
be written using the 4 nearest neighbors of point (x y)
Bilinear interpolation gives much better results than nearest neighbor interpolation with a
modest increase in computational effort
Bicubic interpolation ndash assign for the new (x y) location an intensity that involves the 16
nearest neighbors of the point 3 3
0 0
( ) i ji j
i jv x y c x y
The coefficients cij are obtained solving a 16x16 linear system
intensity levels of the 16 nearest neighbors of 3 3
0 0
( )i ji j
i jc x y x y
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Generally bicubic interpolation does a better job of preserving fine detail than the
bilinear technique Bicubic interpolation is the standard used in commercial image editing
programs such as Adobe Photoshop and Corel Photopaint
Figure 2 (a) is the same as Fig 1 (d) which was obtained by reducing the resolution of
the 1250 dpi in Fig 1(a) to 72 dpi (the size shrank from 3692 times 2812 to 213 times 162) and
then zooming the reduced image back to its original size To generate Fig 1(d) nearest
neighbor interpolation was used (both for shrinking and zooming)
Figures 2(b) and (c) were generated using the same steps but using bilinear and bicubic
interpolation respectively Figures 2(d)+(e)+(f) were obtained by reducing the resolution
from 1250 dpi to 150 dpi (instead of 72 dpi)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Fig 2 ndash Interpolation examples for zooming and shrinking (nearest neighbor linear bicubic)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Neighbors of a Pixel
A pixel p at coordinates (x y) has 4 horizontal and vertical neighbors horizontal vertical( 1 ) ( 1 ) ( 1) ( 1)x y x y x y x y
This set of pixels called the 4-neighbors of p denoted by N4 (p)
The 4 diagonal neighbors of p have coordinates ( 1 1) ( 1 1) ( 1 1) ( 1 1)x y x y x y x y
and are denoted ND(p)
The horizontal vertical and diagonal neighbors are called the 8-neighbors of p denoted
N8 (p)
If (xy) is on the border of the image some of the neighbor locations in ND(p) and N8(p)
fall outside the image
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Adjacency Connectivity Regions Boundaries
Denote by V the set of intensity levels used to define adjacency
- in a binary image V 01 (V=0 V=1)
- in a gray-scale image with 256 possible gray-levels V can be any subset of 0255
We consider 3 types of adjacency
(a) 4-adjacency two pixels p and q with values from V are 4-adjacent if 4( )q N p
(b) 8-adjacency two pixels p and q with values from V are 8-adjacent if 8( )q N p
(c) m-adjacency (mixed adjacency) two pixels p and q with values from V are
m-adjacent if
4( )q N p or
( )Dq N p and the set 4 4( ) ( )N p N q has no pixels whose values are from V
Mixed adjacency is a modification of 8-adjacency It is introduced to eliminate the
ambiguities that often arise when 8-adjacency is used Consider the example
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
binary image
0 1 1 0 1 1 0 1 1
1 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
V
The three pixels at the top (first line) in the above example show multiple (ambiguous)
8-adjacency as indicated by the dashed lines This ambiguity is removed by using
m-adjacency
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
A (digital) path (or curve) from pixel p with coordinates (xy) to q with coordinates (st)
is a sequence of distinct pixels with coordinates
and are adjacent 0 0 1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) 12
n n
i i i i
x y x y x y x y s tx y x y i n
The length of the path is n If 0 0( ) ( )n nx y x y the path is closed
Depending on the type of adjacency considered the paths are 4- 8- or m-paths
Let S denote a subset of pixels in an image Two pixels p and q are said to be connected
in S if there exists a path between them consisting only of pixels from S
S is a connected set if there is a path in S between any 2 pixels in S
Let R be a subset of pixels in an image R is a region of the image if R is a connected set
Two regions R1 and R2 are said to be adjacent if 1 2R R form a connected set Regions
that are not adjacent are said to be disjoint When referring to regions only 4- and
8-adjacency are considered
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Suppose that an image contains K disjoint regions 1 kR k K none of which
touches the image border
the complement of 1
( )K
cu k u u
k
R R R R
We call all the points in Ru the foreground of the image and the points in ( )cuR the
background of the image
The boundary (border or contour) of a region R is the set of points that are adjacent to
points in the complement of R (R)c The border of an image is the set of pixels in the
region that have at least one background neighbor This definition is referred to as the
inner border to distinguish it from the notion of outer border which is the corresponding
border in the background
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Distance measures
For pixels p q and z with coordinates (xy) (st) and (vw) respectively D is a distance
function or metric if
(a) D(p q) ge 0 D(p q) = 0 iff p=q
(b) D(p q) = D(q p)
(c) D(p z) le D(p q) + D(q z)
The Euclidean distance between p and q is defined as 1
2 2 2 22( ) ( ) ( ) ( ) ( )eD p q x s y t x s y t
The pixels q for which ( )eD p q r are the points contained in a disk of radius r
centered at (x y)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
The D4 distance (also called city-block distance) between p and q is defined as
4( ) | | | |D p q x s y t
The pixels q for which 4( )D p q r form a diamond centered at (xy)
4
22 1 2
2 2 1 0 1 22 1 2
2
D
The pixels with D4 = 1 are the 4-neighbors of (x y)
The D8 distance (called the chessboard distance) between p and q is defined as
8( ) max| | | |D p q x s y t
The pixels q for which 8( )D p q r form a square centered at (x y)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
8
2 2 2 2 22 1 1 1 2
2 2 1 0 1 22 1 1 1 22 2 2 2 2
D
The pixels with D8 = 1 are the 8-neighbors of (x y)
D4 and D8 distances are independent of any paths that might exist between p and q
because these distances involve only the coordinates of the point
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Array versus Matrix Operations
An array operation involving one or more images is carried out on a pixel-by-pixel basis
11 12 11 12
21 22 21 22
a a b ba a b b
Array product
11 12 11 12 11 11 12 12
21 22 21 22 21 21 22 21
a a b b a b a ba a b b a b a b
Matrix product
11 12 11 12 11 11 12 21 11 12 12 21
21 22 21 22 21 11 22 21 21 12 22 22
a a b b a b a b a b a ba a b b a b a b a b a b
We assume array operations unless stated otherwise
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Linear versus Nonlinear Operations
One of the most important classifications of image-processing methods is whether it is
linear or nonlinear
( ) ( )H f x y g x y
H is said to be a linear operator if
images1 2 1 2
1 2
( ) ( ) ( ) ( )
H a f x y b f x y a H f x y b H f x y
a b f f
Example of nonlinear operator
the maximum value of the pixels of image max ( )H f f x y f
1 2
0 2 6 5 1 1
2 3 4 7f f a b
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
1 2
0 2 6 5 6 3max max 1 ( 1) max 2
2 3 4 7 2 4a f b f
0 2 6 51 max ( 1) max 3 ( 1)7 4
2 3 4 7
Arithmetic Operations in Image Processing
Let g(xy) denote a corrupted image formed by the addition of noise ( ) ( ) ( )g x y f x y x y
f(xy) ndash the noiseless image η(xy) the noise uncorrelated and has 0 average value
For a random variable z with mean m E[(z-m)2] is the variance (E( ∙ ) is the expected
value) The covariance of two random variables z1 and z2 is defined as E[(z1-m1) (z2-m2)]
The two random variables are uncorrelated when their covariance is 0
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Objective reduce noise by adding a set of noisy images ( )ig x y (technique frequently
used in image enhancement)
1
1( ) ( )K
ii
g x y g x yK
If the noise satisfies the properties stated above we have
2 2( ) ( )
1( ) ( ) g x y x yE g x y f x yK
( ( ))E g x y is the expected value of g and and 2 2( ) ( )g x y x y are the variances of
and g respectively The standard deviation (square root of the variance) at any point in
the average image is
( ) ( )1
g x y x yK
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
As K increases the variability (as measured by the variance or the standard deviation) of
the pixel values at each location (x y) decreases Because ( ) ( )E g x y f x y this
means that ( )g x y approaches f(x y) as the number of noisy images used in the
averaging process increases
An important application of image averaging is in the field of astronomy where imaging
under very low light levels frequently causes sensor noise to render single images
virtually useless for analysis Figure 226(a) shows an 8-bit image in which corruption
was simulated by adding to it Gaussian noise with zero mean and a standard deviation of
64 intensity levels Figures 226(b)-(f) show the result of averaging 5 10 20 50 and 100
images respectively
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Fig 3 Image of Galaxy Pair NGC 3314 corrupted by additive Gaussian noise (left corner) Results of averaging 5 10 20 50
100 noisy images
a b c d e f
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
A frequent application of image subtraction is in the enhancement of differences between
images
(a) (b) (c)
Fig 4 (a) Infrared image of Washington DC area (b) Image obtained from (a) by setting to zero the least
significant bit of each pixel (c) the difference between the two images
Figure 4(b) was obtained by setting to zero the least-significant bit of every pixel in
Figure 4(a) The two images seem almost the same Figure 4(c) is the difference between
images (a) and (b) Black (0) values in Figure (c) indicate locations where there is no
difference between images (a) and (b)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Mask mode radiography ( ) ( ) ( )g x y f x y h x y
h(x y) the mask is an X-ray image of a region of a patientrsquos body captured by an
intensified TV camera (instead of traditional X-ray film) located opposite an X-ray
source The procedure consists of injecting an X-ray contrast medium into the patientrsquos
bloodstream taking a series of images called live images (denoted f(x y)) of the same
anatomical region as h(x y) and subtracting the mask from the series of incoming live
images after injection of the contrast medium
In g(x y) we can find the differences between h and f as enhanced detail
Images being captured at TV rates we obtain a movie showing how the contrast medium
propagates through the various arteries in the area being observed
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
a b c d Fig 5 ndash Angiography ndash subtraction example (a) ndash mask image (b) ndash live image (c) ndash difference between (a) and (b) (d) - image (c) enhanced
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
An important application of image multiplication (and division) is shading correction
Suppose that an imaging sensor produces images in the form ( ) ( ) ( )g x y f x y h x y
f(x y) ndash the ldquoperfect imagerdquo h(x y) ndash the shading function
When the shading function is known
( )( )( )
g x yf x yh x y
h(x y) is unknown but we have access to the imaging system we can obtain an
approximation to the shading function by imaging a target of constant intensity When the
sensor is not available often the shading pattern can be estimated from the image
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
(a) (b) (c)
Fig 6 Shading correction (a) ndash Shaded image of a tungsten filament magnified 130 times (b) - shading pattern (c) corrected image
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Another use of image multiplication is in masking also called region of interest (ROI)
operations The process consists of multiplying a given image by a mask image that has
1rsquos (white) in the ROI and 0rsquos elsewhere There can be more than one ROI in the mask
image and the shape of the ROI can be arbitrary but usually is a rectangular shape
(a) (b) (c)
Fig 7 (a) ndash digital dental X-ray image (b) - ROI mask for teeth with fillings (c) product of (a) and (b)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
In practice most images are displayed using 8 bits the image values are in the range [0255] TIFF JPEG images ndash conversion to this range is automatic The conversion depends on the system used Difference of two images can produce image with values in the range [-255255] Addition of two images ndash range [0510] Many software packages simply set the negative values to 0 and set to 255 all values greater than 255 A more appropriate procedure compute
min( )mf f f
0 ( 255)max( )
ms
m
ff K K K
f
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Spatial Operations
- are performed directly on the pixels of a given image
There are three categories of spatial operations
single-pixel operations
neighborhood operations
geometric spatial transformations
Single-pixel operations
- change the values of intensity for the individual pixels ( )s T z
where z is the intensity of a pixel in the original image and s is the intensity of the
corresponding pixel in the processed image
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Neighborhood operations
Let Sxy denote a set of coordinates of a neighborhood centered on an arbitrary point (x y)
in an image f Neighborhood processing generates new intensity level at point (x y)
based on the values of the intensities of the points in Sxy For example if Sxy is a
rectangular neighborhood of size m x n centered in (x y) we can assign the new value of
intensity by computing the average value of the pixels in Sxy
( )
1( ) ( )xyr c S
g x y f r cm n
The net effect is to perform local blurring in the original image This type of process is
used for example to eliminate small details and thus render ldquoblobsrdquo corresponding to the
largest region of an image
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Geometric spatial transformations and image registration
- modify the spatial relationship between pixels in an image
- these transformations are often called rubber-sheet transformations (analogous to
printing an image on a sheet of rubber and then stretching the sheet according to a
predefined set of rules
A geometric transformation consists of 2 basic operations
1 a spatial transformation of coordinates
2 intensity interpolation that assign intensity values to the spatial transformed
pixels
The coordinate system transformation ( ) [( )]x y T v w
(v w) ndash pixel coordinates in the original image
(x y) ndash pixel coordinates in the transformed image
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
[( )] ( )2 2v wT v w shrinks the original image half its size in both spatial directions
Affine transform
11 1211 21 31
21 2212 22 33
31 32
0[ 1] [ 1] [ 1] 0
1
t tx t v t w t
x y v w T v w t ty t v t w t
t t
(AT)
This transform can scale rotate translate or shear a set of coordinate points depending
on the elements of the matrix T If we want to resize an image rotate it and move the
result to some location we simply form a 3x3 matrix equal to the matrix product of the
scaling rotation and translation matrices from Table 1
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Affine transformations
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
The preceding transformations relocate pixels on an image to new locations To complete
the process we have to assign intensity values to those locations This task is done by
using intensity interpolation (like nearest neighbor bilinear bi-cubic interpolation)
In practice we can use equation (AT) in two basic ways
forward mapping scan the pixels of the input image (v w) compute the new spatial
location (x y) of the corresponding pixel in the new image using (AT) directly
Problems
- intensity assignment when 2 or more pixels in the original image are transformed to
the same location in the output image
- some output locations have no correspondent in the original image (no intensity
assignment)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
inverse mapping scans the output pixel locations and at each location (x y)
computes the corresponding location in the input image (v w) 1( ) ( )v w T x y
It then interpolates among the nearest input pixels to determine the intensity of the output
pixel value
Inverse mappings are more efficient to implement than forward mappings and are used in
numerous commercial implementations of spatial transformations (MATLAB for ex)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Image registration ndash align two or more images of the same scene
In image registration we have available the input and output images but the specific
transformation that produced the output image from the input is generally unknown
The problem is to estimate the transformation function and then use it to register the two
images
- it may be of interest to align (register) two or more image taken at approximately the
same time but using different imaging systems (MRI scanner and a PET scanner)
- align images of a given location taken by the same instrument at different moments
of time (satellite images)
Solving the problem using tie points (also called control points) which are
corresponding points whose locations are known precisely in the input and reference
image
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
How to select tie points
- interactively selecting them
- use of algorithms that try to detect these points
- some imaging systems have physical artifacts (small metallic objects) embedded in
the imaging sensors These objects produce a set of known points (called reseau
marks) directly on all images captured by the system which can be used as guides
for establishing tie points
The problem of estimating the transformation is one of modeling Suppose we have a set
of 4 tie points both on the input image and the reference image A simple model based on
a bilinear approximation is given by
1 2 3 4
5 6 7 8
x c v c w c v w cy c v c w c v w c
(v w) and (x y) are the coordinates of the tie points (we get a 8x8 linear system for ci )
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
When 4 tie points are insufficient to obtain satisfactory registration an approach used
frequently is to select a larger number of tie points and using this new set of tie points
subdivide the image in rectangular regions marked by groups of 4 tie points On the
subregions marked by 4 tie points we applied the transformation model described above
The number of tie points and the sophistication of the model required to solve the register
problem depend on the severity of the geometrical distortion
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
a b c d (a) ndash reference image (b) ndash geometrically distorted image (c) - registered image (d) ndash difference between (a) and (c)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Probabilistic Methods
zi = the values of all possible intensities in an MtimesN digital image i=01hellipL-1
p(zk) = the probability that the intensity level zk occurs in the given image
( ) kk
np zM N
nk = the number of times that intensity zk occurs in the image (MN is the total number of
pixels in the image) 1
0( ) 1
L
kk
p z
The mean (average) intensity of an image is given by 1
0( )
L
k kk
m z p z
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
The variance of the intensities is 1
2 2
0( ) ( )
L
k kk
z m p z
The variance is a measure of the spread of the values of z about the mean so it is a
measure of image contrast Usually for measuring image contrast the standard deviation
( ) is used
The n-th moment of a random variable z about the mean is defined as 1
0( ) ( ) ( )
Ln
n k kk
z z m p z
( 20 1 2( ) 1 ( ) 0 ( )z z z )
3( ) 0z the intensities are biased to values higher than the mean
( 3( ) 0z the intensities are biased to values lower than the mean
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
3( ) 0z the intensities are distributed approximately equally on both side of the
mean
Fig1 (a) Low contrast (b) medium contrast (c) high contrast
Figure 1(a) ndash standard deviation 143 (variance = 2045)
Figure 1(b) ndash standard deviation 316 (variance = 9986)
Figure 1(c) ndash standard deviation 492 (variance = 24206)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Intensity Transformations and Spatial Filtering
( ) ( )g x y T f x y
f(x y) ndash input image g(x y) ndash output image T ndash an operator on f defined over a
neighborhood of (x y)
- the neighborhood of the point (x y) Sxy usually is rectangular centered on (x y)
and much smaller in size than the image
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
- spatial filtering the operator T (the neighborhood and the operation applied on it) is
called spatial filter (spatial mask kernel template or window)
( )xyS x y T becomes an intensity (gray-level or mapping) transformation function
( )s T r
s and r are denoting respectively the intensity of g and f at (x y)
Figure 2 left - T produces an output image of higher contrast than the original by
darkening the intensity levels below k and brightening the levels above k ndash this technique
is called contrast stretching
Fig 2 Intensity transformation functions left - contrast stretching right - thresholding function
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Figure 2 right - T produces a binary output image A mapping of this form is called
thresholding function
Some Basic Intensity Transformation Functions
Image Negatives
The negative of an image with intensity levels in [0 L-1] is obtain using the function ( ) 1s T r L r
- equivalent of a photographic negative
- technique suited for enhancing white or gray detail embedded in dark regions of an
image
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Original Negative image
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Log Transformations - constant ( ) log(1 ) 0s T r c r c r
Some basic intensity transformation functions
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
This transformation maps a narrow range of low intensity values in the input into a wider
range An operator of this type is used to expand the values of dark pixels in an image
while compressing the higher-level values The opposite is true for the inverse log
transformation The log functions compress the dynamic range of images with large
variations in pixel values
Figure 4(a) ndash intensity values in the range 0 to 15 x 106
Figure 4(b) = log transformation of Figure 4(a) with c=1 ndash range 0 to 62
a b (a) ndash Fourier spectrum (b) ndash log transformation applied to (a) c=1 Fig 4
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Power-Law (Gamma) Transformations
- positive constants( ) ( ( ) )s T r c r c s c r
Plots of gamma transformation for different values of γ (c=1)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Power-law curves with 1 map a narrow range of dark input values into a wider range
of output values with the opposite being true for higher values of input values The
curves with 1 have the opposite effect of those generated with values of 1
1c - identity transformation
A variety of devices used for image capture printing and display respond according to a
power law The process used to correct these power-law response phenomena is called
gamma correction
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
a b c d (a) ndash aerial image (b) ndash (d) ndash results of applying gamma transformation with c=1 and γ=30 40 and 50 respectively
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Piecewise-Linear Transformations Functions
Contrast stretching
- a process that expands the range of intensity levels in an image so it spans the full
intensity range of the recording tool or display device
a b c d Fig5
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
11
1
2 1 1 21 2
2 1 2 1
22
2
[0 ]
( ) ( )( ) [ ]( ) ( )
( 1 ) [ 1]( 1 )
s r r rrs r r s r rT r r r r
r r r rs L r r r L
L r
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
1 1 2 2r s r s identity transformation (no change)
1 2 1 2 0 1r r s s L thresholding function
Figure 5(b) shows an 8-bit image with low contrast
Figure 5(c) - contrast stretching obtained by setting the parameters 1 1 min 0r s r
2 2 max 1r s r L where rmin and rmax denote the minimum and maximum gray levels
in the image respectively Thus the transformation function stretched the levels linearly
from their original range to the full range [0 L-1]
Figure 5(d) - the thresholding function was used with 1 1 0r s m
2 2 1r s m L where m is the mean gray level in the image
The original image on which these results are based is a scanning electron microscope
image of pollen magnified approximately 700 times
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Intensity-level slicing
- highlighting a specific range of intensities in an image
There are two approaches for intensity-level slicing
1 display in one value (white for example) all the values in the range of interest and in
another (say black) all other intensities (Figure 311 (a))
2 brighten (or darken) the desired range of intensities but leaves unchanged all other
intensities in the image (Figure 311 (b))
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Figure 6 (left) ndash aortic angiogram near the kidney The purpose of intensity slicing is to
highlight the major blood vessels that appear brighter as a result of injecting a contrast
medium Figure 6(middle) shows the result of applying the first technique for a band near
the top of the scale of intensities This type of enhancement produces a binary image
Highlights intensity range [A B] and reduces all other intensities to a lower level
Highlights range [A B] and preserves all other intensities
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
which is useful for studying the shape of the flow of the contrast substance (to detect
blockageshellip)
In Figure 312(right) the second technique was used a band of intensities in the mid-gray
image around the mean intensity was set to black the other intensities remain unchanged
Fig 6 - Aortic angiogram and intensity sliced versions
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Bit-plane slicing
For a 8-bit image f(x y) is a number in [0255] with 8-bit representation in base 2
This technique highlights the contribution made to the whole image appearances by each
of the bits An 8-bit image may be considered as being composed of eight 1-bit planes
(plane 1 ndash the lowest order bit plane 8 ndash the highest order bit)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
Digital Image Processing
Week 1
The binary image for the 8-th bit plane of an 8-bit image can be obtained by processing the input image with a threshold intensity transformation function that maps all the intensities between 0 and 127 to 0 and maps all levels between 128 and 255 to 1 The bit-slicing technique is useful for analyzing the relative importance of each bit in the image ndash helps in determining the proper number of bits to use when quantizing the image The technique is also useful for image compression
- DIP 1 2017
- DIP 02 (2017)
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