DTU Compute Introduction to Medical Image Analysis Plenty of slides adapted from Thomas Moeslunds lectures Rasmus R. Paulsen DTU Compute [email protected] http://www.compute.dtu.dk/courses/02511 http://www.compute.dtu.dk/courses/02512
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DTU Compute
Introduction to Medical Image Analysis
Plenty of slides adapted from Thomas Moeslunds lectures
Rasmus R PaulsenDTU Compute
rapadtudk
httpwwwcomputedtudkcourses02511httpwwwcomputedtudkcourses02512
DTU Compute
2122018Introduction to Medical Image Analysis2 DTU Compute Technical University of Denmark
Lecture 4 ndash Neighbourhood Processing
800 ndash 900 Exercises900 ndash 11ish Lecture
11ish - Exercises
1200 ndash 1300 Lunch break
1300 - Exercises
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
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Point processing
0 2
1 2
1 2 1
2 5 3
1 3
2 2
0 1
1 2 0
2 1 4
1 0 19
12
Input Output
bull The value of the output pixel is only dependent on the value of one input pixel
bull A global operation ndash changes all pixels
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Point processing Grey level enhancement
ndash Process one pixel at a time independent of all other pixelsndash For example used to correct Brightness and Contrast
Too lowcontrastCorrect
Too highcontrast
Too high brightness
Too low brightness
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Neighbourhood processing
0 2
1 2
1 2 1
2 5 3
1 3
2 2
0 1
1 2 0
2 1 4
1 0 19
12
Input Output
bull Several pixels in the input has an effect on the output
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Use of filtering
Noise removal Enhance edges Smoothing
bull Image processing
bull Typically done before actual image analysis
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Salt and pepper noise Pixel values that are very
different from their neighbours
Very bright or very dark spots
Scratches in X-rays
What is that
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Salt and pepper noise Fake example
ndash Let us take a closer look at noise pixels
They are all 0 or 255
Should we just remove all the 0rsquos and 255rsquos from the image
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What is so special about noise What is the value of the pixel
compared to the neighbours Average of the neighbours
ndash 170 Can we compare to the
averagendash Difficult ndash should we remove
all values bigger than average+1
It is difficult to detect noise
172 169 171 168 0 169 172 173 168
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Noise ndash go away We can not tell what pixels
are noise One solution
ndash Set all pixels to the average of the neighbours (and the pixel itself)
Oh nondash Problemsndash The noise ldquopollutesrdquo the
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
149
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What is the median ValueA) 170B) 173C) 169D) 171E) 172
13
0 0 02
A B C D E
169 168 0 170172 173 170 172 170
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The median value The values are sorted from low to high The middle number is picked
ndash The median value
169 168 0 170172 173 170 172 170
0168 169 170 170 170172 172173
Median
Noise has no influence on the median
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Noise away ndash the median filter All pixels are set to the
median of its neighbourhood Noise pixels do not pollute
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
170
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Noise removal ndash average filter
Scanned X-ray with salt and pepper noise
Average filter (3x3)
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Noise removal ndash median filter
Scanned X-ray with salt and pepper noise
Median filter (3x3)
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Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
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Median filterA) 25B) 90C) 198D) 86E) 103
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13
1 0 0
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
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Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
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Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
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Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
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Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
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Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
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Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
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1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
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Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
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1
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14
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0
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
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Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
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Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
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119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
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0
0
0
0
13
Sheet1
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
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Use of smoothing
3x3 7x7 11x11 15x15
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Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
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12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
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Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
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Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
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Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
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Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
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Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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DTU Compute
2122018Introduction to Medical Image Analysis2 DTU Compute Technical University of Denmark
Lecture 4 ndash Neighbourhood Processing
800 ndash 900 Exercises900 ndash 11ish Lecture
11ish - Exercises
1200 ndash 1300 Lunch break
1300 - Exercises
DTU Compute
2122018Introduction to Medical Image Analysis3 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis4 DTU Compute Technical University of Denmark
Point processing
0 2
1 2
1 2 1
2 5 3
1 3
2 2
0 1
1 2 0
2 1 4
1 0 19
12
Input Output
bull The value of the output pixel is only dependent on the value of one input pixel
bull A global operation ndash changes all pixels
DTU Compute
2122018Introduction to Medical Image Analysis5 DTU Compute Technical University of Denmark
Point processing Grey level enhancement
ndash Process one pixel at a time independent of all other pixelsndash For example used to correct Brightness and Contrast
Too lowcontrastCorrect
Too highcontrast
Too high brightness
Too low brightness
DTU Compute
2122018Introduction to Medical Image Analysis6 DTU Compute Technical University of Denmark
Neighbourhood processing
0 2
1 2
1 2 1
2 5 3
1 3
2 2
0 1
1 2 0
2 1 4
1 0 19
12
Input Output
bull Several pixels in the input has an effect on the output
DTU Compute
2122018Introduction to Medical Image Analysis7 DTU Compute Technical University of Denmark
Use of filtering
Noise removal Enhance edges Smoothing
bull Image processing
bull Typically done before actual image analysis
DTU Compute
2122018Introduction to Medical Image Analysis8 DTU Compute Technical University of Denmark
Salt and pepper noise Pixel values that are very
different from their neighbours
Very bright or very dark spots
Scratches in X-rays
What is that
DTU Compute
2122018Introduction to Medical Image Analysis9 DTU Compute Technical University of Denmark
Salt and pepper noise Fake example
ndash Let us take a closer look at noise pixels
They are all 0 or 255
Should we just remove all the 0rsquos and 255rsquos from the image
DTU Compute
2122018Introduction to Medical Image Analysis10 DTU Compute Technical University of Denmark
What is so special about noise What is the value of the pixel
compared to the neighbours Average of the neighbours
ndash 170 Can we compare to the
averagendash Difficult ndash should we remove
all values bigger than average+1
It is difficult to detect noise
172 169 171 168 0 169 172 173 168
DTU Compute
2122018Introduction to Medical Image Analysis11 DTU Compute Technical University of Denmark
Noise ndash go away We can not tell what pixels
are noise One solution
ndash Set all pixels to the average of the neighbours (and the pixel itself)
Oh nondash Problemsndash The noise ldquopollutesrdquo the
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
149
DTU Compute
2122018Introduction to Medical Image Analysis12 DTU Compute Technical University of Denmark
What is the median ValueA) 170B) 173C) 169D) 171E) 172
13
0 0 02
A B C D E
169 168 0 170172 173 170 172 170
Chart1
13
0
0
0
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis13 DTU Compute Technical University of Denmark
The median value The values are sorted from low to high The middle number is picked
ndash The median value
169 168 0 170172 173 170 172 170
0168 169 170 170 170172 172173
Median
Noise has no influence on the median
DTU Compute
2122018Introduction to Medical Image Analysis14 DTU Compute Technical University of Denmark
Noise away ndash the median filter All pixels are set to the
median of its neighbourhood Noise pixels do not pollute
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
170
DTU Compute
2122018Introduction to Medical Image Analysis15 DTU Compute Technical University of Denmark
Noise removal ndash average filter
Scanned X-ray with salt and pepper noise
Average filter (3x3)
DTU Compute
2122018Introduction to Medical Image Analysis16 DTU Compute Technical University of Denmark
Noise removal ndash median filter
Scanned X-ray with salt and pepper noise
Median filter (3x3)
DTU Compute
2122018Introduction to Medical Image Analysis17 DTU Compute Technical University of Denmark
Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
DTU Compute
2122018Introduction to Medical Image Analysis18 DTU Compute Technical University of Denmark
Median filterA) 25B) 90C) 198D) 86E) 103
0
13
1 0 0
A B C D E
Chart1
0
13
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis20 DTU Compute Technical University of Denmark
Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
DTU Compute
2122018Introduction to Medical Image Analysis21 DTU Compute Technical University of Denmark
Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
Chart1
0
12
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis23 DTU Compute Technical University of Denmark
Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
DTU Compute
2122018Introduction to Medical Image Analysis25 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
DTU Compute
2122018Introduction to Medical Image Analysis26 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
DTU Compute
2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
DTU Compute
2122018Introduction to Medical Image Analysis28 DTU Compute Technical University of Denmark
Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
DTU Compute
2122018Introduction to Medical Image Analysis29 DTU Compute Technical University of Denmark
Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
2122018Introduction to Medical Image Analysis30 DTU Compute Technical University of Denmark
Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis31 DTU Compute Technical University of Denmark
Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis34 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis37 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
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DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
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2
0
A B C D E
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DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
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0
0
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DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
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A B C D E
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DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
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DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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DTU Compute
2122018Introduction to Medical Image Analysis3 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis4 DTU Compute Technical University of Denmark
Point processing
0 2
1 2
1 2 1
2 5 3
1 3
2 2
0 1
1 2 0
2 1 4
1 0 19
12
Input Output
bull The value of the output pixel is only dependent on the value of one input pixel
bull A global operation ndash changes all pixels
DTU Compute
2122018Introduction to Medical Image Analysis5 DTU Compute Technical University of Denmark
Point processing Grey level enhancement
ndash Process one pixel at a time independent of all other pixelsndash For example used to correct Brightness and Contrast
Too lowcontrastCorrect
Too highcontrast
Too high brightness
Too low brightness
DTU Compute
2122018Introduction to Medical Image Analysis6 DTU Compute Technical University of Denmark
Neighbourhood processing
0 2
1 2
1 2 1
2 5 3
1 3
2 2
0 1
1 2 0
2 1 4
1 0 19
12
Input Output
bull Several pixels in the input has an effect on the output
DTU Compute
2122018Introduction to Medical Image Analysis7 DTU Compute Technical University of Denmark
Use of filtering
Noise removal Enhance edges Smoothing
bull Image processing
bull Typically done before actual image analysis
DTU Compute
2122018Introduction to Medical Image Analysis8 DTU Compute Technical University of Denmark
Salt and pepper noise Pixel values that are very
different from their neighbours
Very bright or very dark spots
Scratches in X-rays
What is that
DTU Compute
2122018Introduction to Medical Image Analysis9 DTU Compute Technical University of Denmark
Salt and pepper noise Fake example
ndash Let us take a closer look at noise pixels
They are all 0 or 255
Should we just remove all the 0rsquos and 255rsquos from the image
DTU Compute
2122018Introduction to Medical Image Analysis10 DTU Compute Technical University of Denmark
What is so special about noise What is the value of the pixel
compared to the neighbours Average of the neighbours
ndash 170 Can we compare to the
averagendash Difficult ndash should we remove
all values bigger than average+1
It is difficult to detect noise
172 169 171 168 0 169 172 173 168
DTU Compute
2122018Introduction to Medical Image Analysis11 DTU Compute Technical University of Denmark
Noise ndash go away We can not tell what pixels
are noise One solution
ndash Set all pixels to the average of the neighbours (and the pixel itself)
Oh nondash Problemsndash The noise ldquopollutesrdquo the
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
149
DTU Compute
2122018Introduction to Medical Image Analysis12 DTU Compute Technical University of Denmark
What is the median ValueA) 170B) 173C) 169D) 171E) 172
13
0 0 02
A B C D E
169 168 0 170172 173 170 172 170
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0
0
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis13 DTU Compute Technical University of Denmark
The median value The values are sorted from low to high The middle number is picked
ndash The median value
169 168 0 170172 173 170 172 170
0168 169 170 170 170172 172173
Median
Noise has no influence on the median
DTU Compute
2122018Introduction to Medical Image Analysis14 DTU Compute Technical University of Denmark
Noise away ndash the median filter All pixels are set to the
median of its neighbourhood Noise pixels do not pollute
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
170
DTU Compute
2122018Introduction to Medical Image Analysis15 DTU Compute Technical University of Denmark
Noise removal ndash average filter
Scanned X-ray with salt and pepper noise
Average filter (3x3)
DTU Compute
2122018Introduction to Medical Image Analysis16 DTU Compute Technical University of Denmark
Noise removal ndash median filter
Scanned X-ray with salt and pepper noise
Median filter (3x3)
DTU Compute
2122018Introduction to Medical Image Analysis17 DTU Compute Technical University of Denmark
Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
DTU Compute
2122018Introduction to Medical Image Analysis18 DTU Compute Technical University of Denmark
Median filterA) 25B) 90C) 198D) 86E) 103
0
13
1 0 0
A B C D E
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0
13
1
0
0
Sheet1
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2122018Introduction to Medical Image Analysis20 DTU Compute Technical University of Denmark
Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
DTU Compute
2122018Introduction to Medical Image Analysis21 DTU Compute Technical University of Denmark
Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
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0
12
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis23 DTU Compute Technical University of Denmark
Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
DTU Compute
2122018Introduction to Medical Image Analysis25 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
DTU Compute
2122018Introduction to Medical Image Analysis26 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
DTU Compute
2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
DTU Compute
2122018Introduction to Medical Image Analysis28 DTU Compute Technical University of Denmark
Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
DTU Compute
2122018Introduction to Medical Image Analysis29 DTU Compute Technical University of Denmark
Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
2122018Introduction to Medical Image Analysis30 DTU Compute Technical University of Denmark
Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis31 DTU Compute Technical University of Denmark
Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis34 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis37 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
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A B C D E
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2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
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10
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2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
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0
0
2
6
0
Sheet1
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2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
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2
0
A B C D E
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2
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2
0
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DTU Compute
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
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0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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Point processing
0 2
1 2
1 2 1
2 5 3
1 3
2 2
0 1
1 2 0
2 1 4
1 0 19
12
Input Output
bull The value of the output pixel is only dependent on the value of one input pixel
bull A global operation ndash changes all pixels
DTU Compute
2122018Introduction to Medical Image Analysis5 DTU Compute Technical University of Denmark
Point processing Grey level enhancement
ndash Process one pixel at a time independent of all other pixelsndash For example used to correct Brightness and Contrast
Too lowcontrastCorrect
Too highcontrast
Too high brightness
Too low brightness
DTU Compute
2122018Introduction to Medical Image Analysis6 DTU Compute Technical University of Denmark
Neighbourhood processing
0 2
1 2
1 2 1
2 5 3
1 3
2 2
0 1
1 2 0
2 1 4
1 0 19
12
Input Output
bull Several pixels in the input has an effect on the output
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Use of filtering
Noise removal Enhance edges Smoothing
bull Image processing
bull Typically done before actual image analysis
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Salt and pepper noise Pixel values that are very
different from their neighbours
Very bright or very dark spots
Scratches in X-rays
What is that
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Salt and pepper noise Fake example
ndash Let us take a closer look at noise pixels
They are all 0 or 255
Should we just remove all the 0rsquos and 255rsquos from the image
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2122018Introduction to Medical Image Analysis10 DTU Compute Technical University of Denmark
What is so special about noise What is the value of the pixel
compared to the neighbours Average of the neighbours
ndash 170 Can we compare to the
averagendash Difficult ndash should we remove
all values bigger than average+1
It is difficult to detect noise
172 169 171 168 0 169 172 173 168
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2122018Introduction to Medical Image Analysis11 DTU Compute Technical University of Denmark
Noise ndash go away We can not tell what pixels
are noise One solution
ndash Set all pixels to the average of the neighbours (and the pixel itself)
Oh nondash Problemsndash The noise ldquopollutesrdquo the
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
149
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What is the median ValueA) 170B) 173C) 169D) 171E) 172
13
0 0 02
A B C D E
169 168 0 170172 173 170 172 170
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The median value The values are sorted from low to high The middle number is picked
ndash The median value
169 168 0 170172 173 170 172 170
0168 169 170 170 170172 172173
Median
Noise has no influence on the median
DTU Compute
2122018Introduction to Medical Image Analysis14 DTU Compute Technical University of Denmark
Noise away ndash the median filter All pixels are set to the
median of its neighbourhood Noise pixels do not pollute
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
170
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Noise removal ndash average filter
Scanned X-ray with salt and pepper noise
Average filter (3x3)
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Noise removal ndash median filter
Scanned X-ray with salt and pepper noise
Median filter (3x3)
DTU Compute
2122018Introduction to Medical Image Analysis17 DTU Compute Technical University of Denmark
Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
DTU Compute
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Median filterA) 25B) 90C) 198D) 86E) 103
0
13
1 0 0
A B C D E
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0
13
1
0
0
Sheet1
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
DTU Compute
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Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
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0
12
1
0
0
Sheet1
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2122018Introduction to Medical Image Analysis23 DTU Compute Technical University of Denmark
Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
DTU Compute
2122018Introduction to Medical Image Analysis25 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
DTU Compute
2122018Introduction to Medical Image Analysis26 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
DTU Compute
2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
DTU Compute
2122018Introduction to Medical Image Analysis28 DTU Compute Technical University of Denmark
Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
DTU Compute
2122018Introduction to Medical Image Analysis29 DTU Compute Technical University of Denmark
Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
2122018Introduction to Medical Image Analysis30 DTU Compute Technical University of Denmark
Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
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2122018Introduction to Medical Image Analysis34 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
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0
8
1
1
0
Sheet1
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
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2
3
6
1
0
Sheet1
DTU Compute
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| A | |
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| E |
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Point processing Grey level enhancement
ndash Process one pixel at a time independent of all other pixelsndash For example used to correct Brightness and Contrast
Too lowcontrastCorrect
Too highcontrast
Too high brightness
Too low brightness
DTU Compute
2122018Introduction to Medical Image Analysis6 DTU Compute Technical University of Denmark
Neighbourhood processing
0 2
1 2
1 2 1
2 5 3
1 3
2 2
0 1
1 2 0
2 1 4
1 0 19
12
Input Output
bull Several pixels in the input has an effect on the output
DTU Compute
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Use of filtering
Noise removal Enhance edges Smoothing
bull Image processing
bull Typically done before actual image analysis
DTU Compute
2122018Introduction to Medical Image Analysis8 DTU Compute Technical University of Denmark
Salt and pepper noise Pixel values that are very
different from their neighbours
Very bright or very dark spots
Scratches in X-rays
What is that
DTU Compute
2122018Introduction to Medical Image Analysis9 DTU Compute Technical University of Denmark
Salt and pepper noise Fake example
ndash Let us take a closer look at noise pixels
They are all 0 or 255
Should we just remove all the 0rsquos and 255rsquos from the image
DTU Compute
2122018Introduction to Medical Image Analysis10 DTU Compute Technical University of Denmark
What is so special about noise What is the value of the pixel
compared to the neighbours Average of the neighbours
ndash 170 Can we compare to the
averagendash Difficult ndash should we remove
all values bigger than average+1
It is difficult to detect noise
172 169 171 168 0 169 172 173 168
DTU Compute
2122018Introduction to Medical Image Analysis11 DTU Compute Technical University of Denmark
Noise ndash go away We can not tell what pixels
are noise One solution
ndash Set all pixels to the average of the neighbours (and the pixel itself)
Oh nondash Problemsndash The noise ldquopollutesrdquo the
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
149
DTU Compute
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What is the median ValueA) 170B) 173C) 169D) 171E) 172
13
0 0 02
A B C D E
169 168 0 170172 173 170 172 170
Chart1
13
0
0
0
2
Sheet1
DTU Compute
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The median value The values are sorted from low to high The middle number is picked
ndash The median value
169 168 0 170172 173 170 172 170
0168 169 170 170 170172 172173
Median
Noise has no influence on the median
DTU Compute
2122018Introduction to Medical Image Analysis14 DTU Compute Technical University of Denmark
Noise away ndash the median filter All pixels are set to the
median of its neighbourhood Noise pixels do not pollute
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
170
DTU Compute
2122018Introduction to Medical Image Analysis15 DTU Compute Technical University of Denmark
Noise removal ndash average filter
Scanned X-ray with salt and pepper noise
Average filter (3x3)
DTU Compute
2122018Introduction to Medical Image Analysis16 DTU Compute Technical University of Denmark
Noise removal ndash median filter
Scanned X-ray with salt and pepper noise
Median filter (3x3)
DTU Compute
2122018Introduction to Medical Image Analysis17 DTU Compute Technical University of Denmark
Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
DTU Compute
2122018Introduction to Medical Image Analysis18 DTU Compute Technical University of Denmark
Median filterA) 25B) 90C) 198D) 86E) 103
0
13
1 0 0
A B C D E
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0
13
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis20 DTU Compute Technical University of Denmark
Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
DTU Compute
2122018Introduction to Medical Image Analysis21 DTU Compute Technical University of Denmark
Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
Chart1
0
12
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis23 DTU Compute Technical University of Denmark
Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
DTU Compute
2122018Introduction to Medical Image Analysis25 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
DTU Compute
2122018Introduction to Medical Image Analysis26 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
DTU Compute
2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
DTU Compute
2122018Introduction to Medical Image Analysis28 DTU Compute Technical University of Denmark
Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
DTU Compute
2122018Introduction to Medical Image Analysis29 DTU Compute Technical University of Denmark
Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
2122018Introduction to Medical Image Analysis30 DTU Compute Technical University of Denmark
Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis31 DTU Compute Technical University of Denmark
Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis34 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis37 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
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0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
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12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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1
2
2
3
2
Sheet1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
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Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
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Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
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Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
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0
8
1
1
0
Sheet1
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
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2
3
6
1
0
Sheet1
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| B | 12 | ||
| C | 1 | ||
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| E |
| A | 0 | ||
| B | 13 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
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| A | 13 | ||
| B | 0 | ||
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| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
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Neighbourhood processing
0 2
1 2
1 2 1
2 5 3
1 3
2 2
0 1
1 2 0
2 1 4
1 0 19
12
Input Output
bull Several pixels in the input has an effect on the output
DTU Compute
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Use of filtering
Noise removal Enhance edges Smoothing
bull Image processing
bull Typically done before actual image analysis
DTU Compute
2122018Introduction to Medical Image Analysis8 DTU Compute Technical University of Denmark
Salt and pepper noise Pixel values that are very
different from their neighbours
Very bright or very dark spots
Scratches in X-rays
What is that
DTU Compute
2122018Introduction to Medical Image Analysis9 DTU Compute Technical University of Denmark
Salt and pepper noise Fake example
ndash Let us take a closer look at noise pixels
They are all 0 or 255
Should we just remove all the 0rsquos and 255rsquos from the image
DTU Compute
2122018Introduction to Medical Image Analysis10 DTU Compute Technical University of Denmark
What is so special about noise What is the value of the pixel
compared to the neighbours Average of the neighbours
ndash 170 Can we compare to the
averagendash Difficult ndash should we remove
all values bigger than average+1
It is difficult to detect noise
172 169 171 168 0 169 172 173 168
DTU Compute
2122018Introduction to Medical Image Analysis11 DTU Compute Technical University of Denmark
Noise ndash go away We can not tell what pixels
are noise One solution
ndash Set all pixels to the average of the neighbours (and the pixel itself)
Oh nondash Problemsndash The noise ldquopollutesrdquo the
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
149
DTU Compute
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What is the median ValueA) 170B) 173C) 169D) 171E) 172
13
0 0 02
A B C D E
169 168 0 170172 173 170 172 170
Chart1
13
0
0
0
2
Sheet1
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The median value The values are sorted from low to high The middle number is picked
ndash The median value
169 168 0 170172 173 170 172 170
0168 169 170 170 170172 172173
Median
Noise has no influence on the median
DTU Compute
2122018Introduction to Medical Image Analysis14 DTU Compute Technical University of Denmark
Noise away ndash the median filter All pixels are set to the
median of its neighbourhood Noise pixels do not pollute
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
170
DTU Compute
2122018Introduction to Medical Image Analysis15 DTU Compute Technical University of Denmark
Noise removal ndash average filter
Scanned X-ray with salt and pepper noise
Average filter (3x3)
DTU Compute
2122018Introduction to Medical Image Analysis16 DTU Compute Technical University of Denmark
Noise removal ndash median filter
Scanned X-ray with salt and pepper noise
Median filter (3x3)
DTU Compute
2122018Introduction to Medical Image Analysis17 DTU Compute Technical University of Denmark
Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
DTU Compute
2122018Introduction to Medical Image Analysis18 DTU Compute Technical University of Denmark
Median filterA) 25B) 90C) 198D) 86E) 103
0
13
1 0 0
A B C D E
Chart1
0
13
1
0
0
Sheet1
DTU Compute
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
DTU Compute
2122018Introduction to Medical Image Analysis21 DTU Compute Technical University of Denmark
Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
Chart1
0
12
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis23 DTU Compute Technical University of Denmark
Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
DTU Compute
2122018Introduction to Medical Image Analysis25 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
DTU Compute
2122018Introduction to Medical Image Analysis26 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
DTU Compute
2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
DTU Compute
2122018Introduction to Medical Image Analysis28 DTU Compute Technical University of Denmark
Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
DTU Compute
2122018Introduction to Medical Image Analysis29 DTU Compute Technical University of Denmark
Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
2122018Introduction to Medical Image Analysis30 DTU Compute Technical University of Denmark
Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis31 DTU Compute Technical University of Denmark
Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis37 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
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12
0
0
0
0
Sheet1
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2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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1
2
2
3
2
Sheet1
DTU Compute
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
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0
8
1
1
0
Sheet1
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2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
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2
3
6
1
0
Sheet1
DTU Compute
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
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2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| A | |
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| E |
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| D | |
| E |
| A | 12 | ||
| B | 0 | ||
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| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 13 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 13 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 0 | ||
| C | 14 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 12 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 13 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 13 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
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Use of filtering
Noise removal Enhance edges Smoothing
bull Image processing
bull Typically done before actual image analysis
DTU Compute
2122018Introduction to Medical Image Analysis8 DTU Compute Technical University of Denmark
Salt and pepper noise Pixel values that are very
different from their neighbours
Very bright or very dark spots
Scratches in X-rays
What is that
DTU Compute
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Salt and pepper noise Fake example
ndash Let us take a closer look at noise pixels
They are all 0 or 255
Should we just remove all the 0rsquos and 255rsquos from the image
DTU Compute
2122018Introduction to Medical Image Analysis10 DTU Compute Technical University of Denmark
What is so special about noise What is the value of the pixel
compared to the neighbours Average of the neighbours
ndash 170 Can we compare to the
averagendash Difficult ndash should we remove
all values bigger than average+1
It is difficult to detect noise
172 169 171 168 0 169 172 173 168
DTU Compute
2122018Introduction to Medical Image Analysis11 DTU Compute Technical University of Denmark
Noise ndash go away We can not tell what pixels
are noise One solution
ndash Set all pixels to the average of the neighbours (and the pixel itself)
Oh nondash Problemsndash The noise ldquopollutesrdquo the
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
149
DTU Compute
2122018Introduction to Medical Image Analysis12 DTU Compute Technical University of Denmark
What is the median ValueA) 170B) 173C) 169D) 171E) 172
13
0 0 02
A B C D E
169 168 0 170172 173 170 172 170
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0
0
0
2
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The median value The values are sorted from low to high The middle number is picked
ndash The median value
169 168 0 170172 173 170 172 170
0168 169 170 170 170172 172173
Median
Noise has no influence on the median
DTU Compute
2122018Introduction to Medical Image Analysis14 DTU Compute Technical University of Denmark
Noise away ndash the median filter All pixels are set to the
median of its neighbourhood Noise pixels do not pollute
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
170
DTU Compute
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Noise removal ndash average filter
Scanned X-ray with salt and pepper noise
Average filter (3x3)
DTU Compute
2122018Introduction to Medical Image Analysis16 DTU Compute Technical University of Denmark
Noise removal ndash median filter
Scanned X-ray with salt and pepper noise
Median filter (3x3)
DTU Compute
2122018Introduction to Medical Image Analysis17 DTU Compute Technical University of Denmark
Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
DTU Compute
2122018Introduction to Medical Image Analysis18 DTU Compute Technical University of Denmark
Median filterA) 25B) 90C) 198D) 86E) 103
0
13
1 0 0
A B C D E
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0
13
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis20 DTU Compute Technical University of Denmark
Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
DTU Compute
2122018Introduction to Medical Image Analysis21 DTU Compute Technical University of Denmark
Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
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0
12
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis23 DTU Compute Technical University of Denmark
Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
DTU Compute
2122018Introduction to Medical Image Analysis25 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
DTU Compute
2122018Introduction to Medical Image Analysis26 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
DTU Compute
2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
DTU Compute
2122018Introduction to Medical Image Analysis28 DTU Compute Technical University of Denmark
Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
DTU Compute
2122018Introduction to Medical Image Analysis29 DTU Compute Technical University of Denmark
Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
2122018Introduction to Medical Image Analysis30 DTU Compute Technical University of Denmark
Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis31 DTU Compute Technical University of Denmark
Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis37 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
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Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
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| E |
| A | 12 | ||
| B | 0 | ||
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| A | |
| B | |
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| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 13 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 13 | ||
| B | 0 | ||
| C | 0 | ||
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| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 0 | ||
| C | 14 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 12 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 13 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 13 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis8 DTU Compute Technical University of Denmark
Salt and pepper noise Pixel values that are very
different from their neighbours
Very bright or very dark spots
Scratches in X-rays
What is that
DTU Compute
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Salt and pepper noise Fake example
ndash Let us take a closer look at noise pixels
They are all 0 or 255
Should we just remove all the 0rsquos and 255rsquos from the image
DTU Compute
2122018Introduction to Medical Image Analysis10 DTU Compute Technical University of Denmark
What is so special about noise What is the value of the pixel
compared to the neighbours Average of the neighbours
ndash 170 Can we compare to the
averagendash Difficult ndash should we remove
all values bigger than average+1
It is difficult to detect noise
172 169 171 168 0 169 172 173 168
DTU Compute
2122018Introduction to Medical Image Analysis11 DTU Compute Technical University of Denmark
Noise ndash go away We can not tell what pixels
are noise One solution
ndash Set all pixels to the average of the neighbours (and the pixel itself)
Oh nondash Problemsndash The noise ldquopollutesrdquo the
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
149
DTU Compute
2122018Introduction to Medical Image Analysis12 DTU Compute Technical University of Denmark
What is the median ValueA) 170B) 173C) 169D) 171E) 172
13
0 0 02
A B C D E
169 168 0 170172 173 170 172 170
Chart1
13
0
0
0
2
Sheet1
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The median value The values are sorted from low to high The middle number is picked
ndash The median value
169 168 0 170172 173 170 172 170
0168 169 170 170 170172 172173
Median
Noise has no influence on the median
DTU Compute
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Noise away ndash the median filter All pixels are set to the
median of its neighbourhood Noise pixels do not pollute
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
170
DTU Compute
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Noise removal ndash average filter
Scanned X-ray with salt and pepper noise
Average filter (3x3)
DTU Compute
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Noise removal ndash median filter
Scanned X-ray with salt and pepper noise
Median filter (3x3)
DTU Compute
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Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
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Median filterA) 25B) 90C) 198D) 86E) 103
0
13
1 0 0
A B C D E
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0
13
1
0
0
Sheet1
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
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Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
Chart1
0
12
1
0
0
Sheet1
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
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Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
DTU Compute
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
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Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
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Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
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Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
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Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
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1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
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Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
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Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
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Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
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119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
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Use of smoothing
3x3 7x7 11x11 15x15
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Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
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0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
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Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
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Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
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Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
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Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
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2
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
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Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
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Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
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0 0
A B C D E
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0
0
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0
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Sheet1
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
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0
0
2
6
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Sheet1
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
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2
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0
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
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0
0
1
4
1
Sheet1
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
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0 0
A B C D E
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0
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Sheet1
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
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0
0
5
0
6
Sheet1
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
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Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| D | |
| E |
| A | 13 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 0 | ||
| C | 14 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 12 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 13 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 13 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
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Salt and pepper noise Fake example
ndash Let us take a closer look at noise pixels
They are all 0 or 255
Should we just remove all the 0rsquos and 255rsquos from the image
DTU Compute
2122018Introduction to Medical Image Analysis10 DTU Compute Technical University of Denmark
What is so special about noise What is the value of the pixel
compared to the neighbours Average of the neighbours
ndash 170 Can we compare to the
averagendash Difficult ndash should we remove
all values bigger than average+1
It is difficult to detect noise
172 169 171 168 0 169 172 173 168
DTU Compute
2122018Introduction to Medical Image Analysis11 DTU Compute Technical University of Denmark
Noise ndash go away We can not tell what pixels
are noise One solution
ndash Set all pixels to the average of the neighbours (and the pixel itself)
Oh nondash Problemsndash The noise ldquopollutesrdquo the
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
149
DTU Compute
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What is the median ValueA) 170B) 173C) 169D) 171E) 172
13
0 0 02
A B C D E
169 168 0 170172 173 170 172 170
Chart1
13
0
0
0
2
Sheet1
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The median value The values are sorted from low to high The middle number is picked
ndash The median value
169 168 0 170172 173 170 172 170
0168 169 170 170 170172 172173
Median
Noise has no influence on the median
DTU Compute
2122018Introduction to Medical Image Analysis14 DTU Compute Technical University of Denmark
Noise away ndash the median filter All pixels are set to the
median of its neighbourhood Noise pixels do not pollute
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
170
DTU Compute
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Noise removal ndash average filter
Scanned X-ray with salt and pepper noise
Average filter (3x3)
DTU Compute
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Noise removal ndash median filter
Scanned X-ray with salt and pepper noise
Median filter (3x3)
DTU Compute
2122018Introduction to Medical Image Analysis17 DTU Compute Technical University of Denmark
Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
DTU Compute
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Median filterA) 25B) 90C) 198D) 86E) 103
0
13
1 0 0
A B C D E
Chart1
0
13
1
0
0
Sheet1
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
DTU Compute
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Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
Chart1
0
12
1
0
0
Sheet1
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
DTU Compute
2122018Introduction to Medical Image Analysis25 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
DTU Compute
2122018Introduction to Medical Image Analysis26 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
DTU Compute
2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
DTU Compute
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Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
DTU Compute
2122018Introduction to Medical Image Analysis29 DTU Compute Technical University of Denmark
Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
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Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
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Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
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Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
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119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
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2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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1
2
2
3
2
Sheet1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
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2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
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2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
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| E | 6 |
| A | |
| B | |
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| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 13 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 12 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 13 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 13 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 0 | ||
| C | 14 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 12 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 13 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 13 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis10 DTU Compute Technical University of Denmark
What is so special about noise What is the value of the pixel
compared to the neighbours Average of the neighbours
ndash 170 Can we compare to the
averagendash Difficult ndash should we remove
all values bigger than average+1
It is difficult to detect noise
172 169 171 168 0 169 172 173 168
DTU Compute
2122018Introduction to Medical Image Analysis11 DTU Compute Technical University of Denmark
Noise ndash go away We can not tell what pixels
are noise One solution
ndash Set all pixels to the average of the neighbours (and the pixel itself)
Oh nondash Problemsndash The noise ldquopollutesrdquo the
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
149
DTU Compute
2122018Introduction to Medical Image Analysis12 DTU Compute Technical University of Denmark
What is the median ValueA) 170B) 173C) 169D) 171E) 172
13
0 0 02
A B C D E
169 168 0 170172 173 170 172 170
Chart1
13
0
0
0
2
Sheet1
DTU Compute
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The median value The values are sorted from low to high The middle number is picked
ndash The median value
169 168 0 170172 173 170 172 170
0168 169 170 170 170172 172173
Median
Noise has no influence on the median
DTU Compute
2122018Introduction to Medical Image Analysis14 DTU Compute Technical University of Denmark
Noise away ndash the median filter All pixels are set to the
median of its neighbourhood Noise pixels do not pollute
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
170
DTU Compute
2122018Introduction to Medical Image Analysis15 DTU Compute Technical University of Denmark
Noise removal ndash average filter
Scanned X-ray with salt and pepper noise
Average filter (3x3)
DTU Compute
2122018Introduction to Medical Image Analysis16 DTU Compute Technical University of Denmark
Noise removal ndash median filter
Scanned X-ray with salt and pepper noise
Median filter (3x3)
DTU Compute
2122018Introduction to Medical Image Analysis17 DTU Compute Technical University of Denmark
Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
DTU Compute
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Median filterA) 25B) 90C) 198D) 86E) 103
0
13
1 0 0
A B C D E
Chart1
0
13
1
0
0
Sheet1
DTU Compute
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
DTU Compute
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Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
Chart1
0
12
1
0
0
Sheet1
DTU Compute
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
DTU Compute
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
DTU Compute
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
DTU Compute
2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
DTU Compute
2122018Introduction to Medical Image Analysis28 DTU Compute Technical University of Denmark
Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
DTU Compute
2122018Introduction to Medical Image Analysis29 DTU Compute Technical University of Denmark
Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
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Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
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Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
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Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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DTU Compute
2122018Introduction to Medical Image Analysis11 DTU Compute Technical University of Denmark
Noise ndash go away We can not tell what pixels
are noise One solution
ndash Set all pixels to the average of the neighbours (and the pixel itself)
Oh nondash Problemsndash The noise ldquopollutesrdquo the
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
149
DTU Compute
2122018Introduction to Medical Image Analysis12 DTU Compute Technical University of Denmark
What is the median ValueA) 170B) 173C) 169D) 171E) 172
13
0 0 02
A B C D E
169 168 0 170172 173 170 172 170
Chart1
13
0
0
0
2
Sheet1
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The median value The values are sorted from low to high The middle number is picked
ndash The median value
169 168 0 170172 173 170 172 170
0168 169 170 170 170172 172173
Median
Noise has no influence on the median
DTU Compute
2122018Introduction to Medical Image Analysis14 DTU Compute Technical University of Denmark
Noise away ndash the median filter All pixels are set to the
median of its neighbourhood Noise pixels do not pollute
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
170
DTU Compute
2122018Introduction to Medical Image Analysis15 DTU Compute Technical University of Denmark
Noise removal ndash average filter
Scanned X-ray with salt and pepper noise
Average filter (3x3)
DTU Compute
2122018Introduction to Medical Image Analysis16 DTU Compute Technical University of Denmark
Noise removal ndash median filter
Scanned X-ray with salt and pepper noise
Median filter (3x3)
DTU Compute
2122018Introduction to Medical Image Analysis17 DTU Compute Technical University of Denmark
Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
DTU Compute
2122018Introduction to Medical Image Analysis18 DTU Compute Technical University of Denmark
Median filterA) 25B) 90C) 198D) 86E) 103
0
13
1 0 0
A B C D E
Chart1
0
13
1
0
0
Sheet1
DTU Compute
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
DTU Compute
2122018Introduction to Medical Image Analysis21 DTU Compute Technical University of Denmark
Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
Chart1
0
12
1
0
0
Sheet1
DTU Compute
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
DTU Compute
2122018Introduction to Medical Image Analysis25 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
DTU Compute
2122018Introduction to Medical Image Analysis26 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
DTU Compute
2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
DTU Compute
2122018Introduction to Medical Image Analysis28 DTU Compute Technical University of Denmark
Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
DTU Compute
2122018Introduction to Medical Image Analysis29 DTU Compute Technical University of Denmark
Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
2122018Introduction to Medical Image Analysis30 DTU Compute Technical University of Denmark
Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis31 DTU Compute Technical University of Denmark
Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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What is the median ValueA) 170B) 173C) 169D) 171E) 172
13
0 0 02
A B C D E
169 168 0 170172 173 170 172 170
Chart1
13
0
0
0
2
Sheet1
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The median value The values are sorted from low to high The middle number is picked
ndash The median value
169 168 0 170172 173 170 172 170
0168 169 170 170 170172 172173
Median
Noise has no influence on the median
DTU Compute
2122018Introduction to Medical Image Analysis14 DTU Compute Technical University of Denmark
Noise away ndash the median filter All pixels are set to the
median of its neighbourhood Noise pixels do not pollute
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
170
DTU Compute
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Noise removal ndash average filter
Scanned X-ray with salt and pepper noise
Average filter (3x3)
DTU Compute
2122018Introduction to Medical Image Analysis16 DTU Compute Technical University of Denmark
Noise removal ndash median filter
Scanned X-ray with salt and pepper noise
Median filter (3x3)
DTU Compute
2122018Introduction to Medical Image Analysis17 DTU Compute Technical University of Denmark
Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
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Median filterA) 25B) 90C) 198D) 86E) 103
0
13
1 0 0
A B C D E
Chart1
0
13
1
0
0
Sheet1
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
DTU Compute
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Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
Chart1
0
12
1
0
0
Sheet1
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
DTU Compute
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
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Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
DTU Compute
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Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
DTU Compute
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Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
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Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
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Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
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Use of smoothing
3x3 7x7 11x11 15x15
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Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
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Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
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Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
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Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
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Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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2
2
3
2
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
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2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
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Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
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0
0
2
6
0
Sheet1
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
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2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
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2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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The median value The values are sorted from low to high The middle number is picked
ndash The median value
169 168 0 170172 173 170 172 170
0168 169 170 170 170172 172173
Median
Noise has no influence on the median
DTU Compute
2122018Introduction to Medical Image Analysis14 DTU Compute Technical University of Denmark
Noise away ndash the median filter All pixels are set to the
median of its neighbourhood Noise pixels do not pollute
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
170
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Noise removal ndash average filter
Scanned X-ray with salt and pepper noise
Average filter (3x3)
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Noise removal ndash median filter
Scanned X-ray with salt and pepper noise
Median filter (3x3)
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Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
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Median filterA) 25B) 90C) 198D) 86E) 103
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
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Median filter
A) 3B) 84C) 112D) 73E) 202
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
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Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
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2122018Introduction to Medical Image Analysis26 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
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2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
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Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
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Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
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2122018Introduction to Medical Image Analysis30 DTU Compute Technical University of Denmark
Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
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1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
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2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
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14
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
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Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
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119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
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0
0
0
13
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
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Use of smoothing
3x3 7x7 11x11 15x15
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Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
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0
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
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Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
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Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
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Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
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Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
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2
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
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Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
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Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
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2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
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A B C D E
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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Sheet1
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2122018Introduction to Medical Image Analysis13 DTU Compute Technical University of Denmark
The median value The values are sorted from low to high The middle number is picked
ndash The median value
169 168 0 170172 173 170 172 170
0168 169 170 170 170172 172173
Median
Noise has no influence on the median
DTU Compute
2122018Introduction to Medical Image Analysis14 DTU Compute Technical University of Denmark
Noise away ndash the median filter All pixels are set to the
median of its neighbourhood Noise pixels do not pollute
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
170
DTU Compute
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Noise removal ndash average filter
Scanned X-ray with salt and pepper noise
Average filter (3x3)
DTU Compute
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Noise removal ndash median filter
Scanned X-ray with salt and pepper noise
Median filter (3x3)
DTU Compute
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Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
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Median filterA) 25B) 90C) 198D) 86E) 103
0
13
1 0 0
A B C D E
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Sheet1
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
DTU Compute
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Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
Chart1
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12
1
0
0
Sheet1
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
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2122018Introduction to Medical Image Analysis26 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
DTU Compute
2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
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2122018Introduction to Medical Image Analysis28 DTU Compute Technical University of Denmark
Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
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2122018Introduction to Medical Image Analysis29 DTU Compute Technical University of Denmark
Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
2122018Introduction to Medical Image Analysis30 DTU Compute Technical University of Denmark
Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis31 DTU Compute Technical University of Denmark
Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
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2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
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Use of smoothing
3x3 7x7 11x11 15x15
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Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
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Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
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Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
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Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
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Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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1
2
2
3
2
Sheet1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
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Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
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Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
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A B C D E
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
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Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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The median value The values are sorted from low to high The middle number is picked
ndash The median value
169 168 0 170172 173 170 172 170
0168 169 170 170 170172 172173
Median
Noise has no influence on the median
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Noise away ndash the median filter All pixels are set to the
median of its neighbourhood Noise pixels do not pollute
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
170
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Noise removal ndash average filter
Scanned X-ray with salt and pepper noise
Average filter (3x3)
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Noise removal ndash median filter
Scanned X-ray with salt and pepper noise
Median filter (3x3)
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Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
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Median filterA) 25B) 90C) 198D) 86E) 103
0
13
1 0 0
A B C D E
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Sheet1
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
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Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
Chart1
0
12
1
0
0
Sheet1
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
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Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
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Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
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Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
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Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
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Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
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1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
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Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
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Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
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Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
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119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
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Use of smoothing
3x3 7x7 11x11 15x15
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Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
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Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
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Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
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Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
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Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
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Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
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A B C D E
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
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0
6
A B C D E
Chart1
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0
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
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2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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Noise away ndash the median filter All pixels are set to the
median of its neighbourhood Noise pixels do not pollute
good pixels
172 169 171 168 0 169 172 173 168
170
169 168 0 170 172173 170 172 170
170
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Noise removal ndash average filter
Scanned X-ray with salt and pepper noise
Average filter (3x3)
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Noise removal ndash median filter
Scanned X-ray with salt and pepper noise
Median filter (3x3)
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Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
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Median filterA) 25B) 90C) 198D) 86E) 103
0
13
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A B C D E
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
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Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
Chart1
0
12
1
0
0
Sheet1
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
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Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
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2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
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Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
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Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
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2122018Introduction to Medical Image Analysis30 DTU Compute Technical University of Denmark
Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
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1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
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2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
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Use of smoothing
3x3 7x7 11x11 15x15
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Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
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Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
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Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
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Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
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Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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1
2
2
3
2
Sheet1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
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Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
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2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
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A B C D E
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
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0
6
A B C D E
Chart1
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| E |
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Noise removal ndash average filter
Scanned X-ray with salt and pepper noise
Average filter (3x3)
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Noise removal ndash median filter
Scanned X-ray with salt and pepper noise
Median filter (3x3)
DTU Compute
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Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
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Median filterA) 25B) 90C) 198D) 86E) 103
0
13
1 0 0
A B C D E
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13
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0
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
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Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
Chart1
0
12
1
0
0
Sheet1
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
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Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
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Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
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Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
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Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
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Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
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Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
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Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
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119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
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Use of smoothing
3x3 7x7 11x11 15x15
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Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
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Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
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Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
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Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
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Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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1
2
2
3
2
Sheet1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
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A B C D E
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
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0
1
4
1
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
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0
3
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0
Sheet1
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
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0
6
A B C D E
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0
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| B | 13 | ||
| C | 1 | ||
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Noise removal ndash median filter
Scanned X-ray with salt and pepper noise
Median filter (3x3)
DTU Compute
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Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
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Median filterA) 25B) 90C) 198D) 86E) 103
0
13
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A B C D E
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0
13
1
0
0
Sheet1
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
DTU Compute
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Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
Chart1
0
12
1
0
0
Sheet1
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
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Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
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Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
DTU Compute
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Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
DTU Compute
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Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
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Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
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Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
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Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
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Use of smoothing
3x3 7x7 11x11 15x15
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Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
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Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
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Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
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Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
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Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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1
2
2
3
2
Sheet1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
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2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
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A B C D E
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
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1
4
1
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
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0
3
0
0
Sheet1
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
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0
6
A B C D E
Chart1
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0
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| B | 13 | ||
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Image Filtering Creates a new filtered image Output pixel is computed based
on a neighbourhood in the input image
3 x 3 neighbourhoodndash Filter size 3 x 3ndash Kernel size 3 x 3ndash Mask size 3 x 3
Larger filters often usedndash Size
7 x 7ndash Number of elements
49
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Median filterA) 25B) 90C) 198D) 86E) 103
0
13
1 0 0
A B C D E
Chart1
0
13
1
0
0
Sheet1
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
DTU Compute
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Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
Chart1
0
12
1
0
0
Sheet1
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
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Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
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Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
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Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
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Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
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Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
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Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
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Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
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119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
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Use of smoothing
3x3 7x7 11x11 15x15
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Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
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Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
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Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
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Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
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Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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1
2
2
3
2
Sheet1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
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2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
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A B C D E
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
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6
A B C D E
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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Median filterA) 25B) 90C) 198D) 86E) 103
0
13
1 0 0
A B C D E
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13
1
0
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Sheet1
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
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Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
10 0
A B C D E
Chart1
0
12
1
0
0
Sheet1
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
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Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
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Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
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Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
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Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
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1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
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Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
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Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
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119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
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Use of smoothing
3x3 7x7 11x11 15x15
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Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
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Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
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Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
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Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
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Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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2
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
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A B C D E
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
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A B C D E
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
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0
6
A B C D E
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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13
1
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
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Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
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A B C D E
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12
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
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Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
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Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
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Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
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Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
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Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
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Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
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Use of smoothing
3x3 7x7 11x11 15x15
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Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
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12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
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Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
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Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
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Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
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Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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2
2
3
2
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
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Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
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Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
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6
A B C D E
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
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Median filter
A) 3B) 84C) 112D) 73E) 202
0
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A B C D E
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
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2122018Introduction to Medical Image Analysis26 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
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Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
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Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
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Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
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Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis31 DTU Compute Technical University of Denmark
Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
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1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
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Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
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Use of smoothing
3x3 7x7 11x11 15x15
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Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
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Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
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Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
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Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
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Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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1
2
2
3
2
Sheet1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
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Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
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2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
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2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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Rank filters Based on sorting the pixel
values in the neighbouring region
Minimum rank filterndash Darker image Noise problems
Maximum rank filterndash Lighter image Noise problems
Difference filterndash Enhances changes (edges)
0168 169 170 170 170172 172173
-
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Median filter
A) 3B) 84C) 112D) 73E) 202
0
12
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A B C D E
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
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Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
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Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
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Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
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Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
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1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
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Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
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Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
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Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
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119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
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Use of smoothing
3x3 7x7 11x11 15x15
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Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
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Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
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Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
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Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
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Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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1
2
2
3
2
Sheet1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
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Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
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Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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Median filter
A) 3B) 84C) 112D) 73E) 202
0
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
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2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
DTU Compute
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Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
DTU Compute
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Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
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Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
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Use of smoothing
3x3 7x7 11x11 15x15
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Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
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0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
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Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
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Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
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Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
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Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
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Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
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A B C D E
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0
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23
6
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A B C D E
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2
3
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0
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
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0
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
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2122018Introduction to Medical Image Analysis25 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
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2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
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2122018Introduction to Medical Image Analysis28 DTU Compute Technical University of Denmark
Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
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Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
2122018Introduction to Medical Image Analysis30 DTU Compute Technical University of Denmark
Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
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1
0
14
0
0
Sheet1
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
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0
0
0
13
Sheet1
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
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2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
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2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
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2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
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2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
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2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
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Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
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23
6
10
A B C D E
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0
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
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0
0
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
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Sheet1
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2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
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A B C D E
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2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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Sheet1
DTU Compute
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Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
DTU Compute
2122018Introduction to Medical Image Analysis25 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
DTU Compute
2122018Introduction to Medical Image Analysis26 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
DTU Compute
2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
DTU Compute
2122018Introduction to Medical Image Analysis28 DTU Compute Technical University of Denmark
Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
DTU Compute
2122018Introduction to Medical Image Analysis29 DTU Compute Technical University of Denmark
Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
2122018Introduction to Medical Image Analysis30 DTU Compute Technical University of Denmark
Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis31 DTU Compute Technical University of Denmark
Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
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0
0
0
0
13
Sheet1
DTU Compute
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
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0
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0
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
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2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
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2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
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A B C D E
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
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2
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0
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
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0
0
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
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0
0
2
6
0
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
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2
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2
0
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
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0
0
1
4
1
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
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3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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DTU Compute
2122018Introduction to Medical Image Analysis23 DTU Compute Technical University of Denmark
Correlation What is it Two measurements
ndash Low correlationndash High correlation
High correlation means that there is a relation between the values
They look the same Correlation is a measure of
similarity
Low
High
Muscles
Mat
h Ski
lls
Muscles
Ben
ch P
ress
DTU Compute
2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
DTU Compute
2122018Introduction to Medical Image Analysis25 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
DTU Compute
2122018Introduction to Medical Image Analysis26 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
DTU Compute
2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
DTU Compute
2122018Introduction to Medical Image Analysis28 DTU Compute Technical University of Denmark
Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
DTU Compute
2122018Introduction to Medical Image Analysis29 DTU Compute Technical University of Denmark
Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
2122018Introduction to Medical Image Analysis30 DTU Compute Technical University of Denmark
Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis31 DTU Compute Technical University of Denmark
Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
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Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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2122018Introduction to Medical Image Analysis24 DTU Compute Technical University of Denmark
Why do we need similarity Image analysis is also about recognition of patterns Often an example pattern is used We need something to tell us if there is a high match
between our pattern and a part of the image
Example pattern
Image
Find matches
DTU Compute
2122018Introduction to Medical Image Analysis25 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
DTU Compute
2122018Introduction to Medical Image Analysis26 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
DTU Compute
2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
DTU Compute
2122018Introduction to Medical Image Analysis28 DTU Compute Technical University of Denmark
Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
DTU Compute
2122018Introduction to Medical Image Analysis29 DTU Compute Technical University of Denmark
Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
2122018Introduction to Medical Image Analysis30 DTU Compute Technical University of Denmark
Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis31 DTU Compute Technical University of Denmark
Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
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2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
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12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
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2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
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8
1
1
0
Sheet1
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
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2
3
6
1
0
Sheet1
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 13 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 12 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 13 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 13 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 0 | ||
| C | 14 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis25 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise
Kernel
Signal (1D image)
41
DTU Compute
2122018Introduction to Medical Image Analysis26 DTU Compute Technical University of Denmark
Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
DTU Compute
2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
DTU Compute
2122018Introduction to Medical Image Analysis28 DTU Compute Technical University of Denmark
Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
DTU Compute
2122018Introduction to Medical Image Analysis29 DTU Compute Technical University of Denmark
Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
2122018Introduction to Medical Image Analysis30 DTU Compute Technical University of Denmark
Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis31 DTU Compute Technical University of Denmark
Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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1
2
2
3
2
Sheet1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
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0
8
1
1
0
Sheet1
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| A | 12 | ||
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| E | 13 |
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| B | 0 | ||
| C | 14 | ||
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| A | |
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| C | |
| D | |
| E |
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Correlation (1D)
1 1 2 2 1 1 2 2 1 1
1 2 1
45
47
47
45
45
47
47
45
41
DTU Compute
2122018Introduction to Medical Image Analysis27 DTU Compute Technical University of Denmark
Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
DTU Compute
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Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
DTU Compute
2122018Introduction to Medical Image Analysis29 DTU Compute Technical University of Denmark
Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
2122018Introduction to Medical Image Analysis30 DTU Compute Technical University of Denmark
Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis31 DTU Compute Technical University of Denmark
Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis37 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
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2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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1
2
2
3
2
Sheet1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
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2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
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Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
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0
8
1
1
0
Sheet1
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
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2
3
6
1
0
Sheet1
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
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2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| A | 13 | ||
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| A | 1 | ||
| B | 0 | ||
| C | 14 | ||
| D | 0 | ||
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| A | |
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| C | |
| D | |
| E |
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Normalisation
1 1 2 2 1 1 2 2 1 1
1 2 1
45
41
1 1 + 2 1 + 1 2 = 5
Normalise41
bull The sum of the kernel elements is used
bull Keep the values in the same range as the input image
Sum is 4
DTU Compute
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Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
DTU Compute
2122018Introduction to Medical Image Analysis29 DTU Compute Technical University of Denmark
Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
2122018Introduction to Medical Image Analysis30 DTU Compute Technical University of Denmark
Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis31 DTU Compute Technical University of Denmark
Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis37 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
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Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
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119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
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Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
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Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
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2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
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Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
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Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
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2
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
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Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
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Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
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Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
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0
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
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1
0
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
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0
0
2
6
0
Sheet1
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
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2
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2
0
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| A | 12 | ||
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| E |
| A | 0 | ||
| B | 0 | ||
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| E | 13 |
| A | |
| B | |
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| A | 13 | ||
| B | 0 | ||
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| A | |
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| E |
| A | 1 | ||
| B | 0 | ||
| C | 14 | ||
| D | 0 | ||
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| A | |
| B | |
| C | |
| D | |
| E |
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Normalisation Normalisation factor
ndash Sum of kernel coefficients
1 2 1h(x)
119909119909
ℎ 119909119909 = 1 + 2 + 1
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Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
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Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
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2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
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2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
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2
2
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2
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
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8
1
1
0
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
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1
0
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
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0
0
1
0
0
Sheet1
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
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0
0
2
6
0
Sheet1
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
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2
1
3
2
0
Sheet1
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| E |
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| C | 0 | ||
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| E | 13 |
| A | |
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| E |
| A | 13 | ||
| B | 0 | ||
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| E |
| A | 1 | ||
| B | 0 | ||
| C | 14 | ||
| D | 0 | ||
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| A | |
| B | |
| C | |
| D | |
| E |
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Correlation on images The filter is now 2D 1 1 1
1 1 1
1 1 1
0 2 4
1 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
912
91
Input OutputKernel
Kernel coefficients
DTU Compute
2122018Introduction to Medical Image Analysis30 DTU Compute Technical University of Denmark
Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis31 DTU Compute Technical University of Denmark
Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
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Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
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119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
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2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
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8
1
1
0
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
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1
0
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
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0
0
1
0
0
Sheet1
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
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0
0
2
6
0
Sheet1
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
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2
1
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2
0
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
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2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| E | 13 |
| A | |
| B | |
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| E |
| A | 13 | ||
| B | 0 | ||
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| B | 0 | ||
| C | 14 | ||
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| D | |
| E |
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Correlation on images
1 1 11 1 11 1 1
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
912
911
91
Input Output
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
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Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
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A B C D E
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
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A B C D E
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Sheet1
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
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6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
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A B C D E
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
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A B C D E
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
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0 0
A B C D E
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2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
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A B C D E
Chart1
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What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
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Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| E |
DTU Compute
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Correlation on images
Input Output
The mask is moved row by row
No values at the border
0 2 4
1 2 2
1 6 3
1 2 0
2 1 4
1 0 1
1 3 1
2 2 2
0 1 3
1 2 12 5 3
2 1 3
DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
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2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
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2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
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0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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1
2
2
3
2
Sheet1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
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2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
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2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
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0
8
1
1
0
Sheet1
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
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1
0
Sheet1
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
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2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
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Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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DTU Compute
2122018Introduction to Medical Image Analysis32 DTU Compute Technical University of Denmark
1 2 1
1 3 1
1 2 1
-1 -2 -1
0 0 0
1 2 1
Draw your filter
DTU Compute
2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis37 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
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0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
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0
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Sheet1
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
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2
0
A B C D E
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
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0
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
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2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| E | 0 |
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| B | 8 | ||
| C | 1 | ||
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| C | 0 | ||
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| A | 12 | ||
| B | 0 | ||
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| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 13 |
| A | |
| B | |
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| D | |
| E |
| A | 13 | ||
| B | 0 | ||
| C | 0 | ||
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| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 0 | ||
| C | 14 | ||
| D | 0 | ||
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| A | |
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| E |
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2122018Introduction to Medical Image Analysis33 DTU Compute Technical University of Denmark
Correlation on images ndash no normalisationA) 4B) 7C) 16D) 23E) 25
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 11 0
14
0 0
A B C D E
Chart1
1
0
14
0
0
Sheet1
DTU Compute
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis37 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
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2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
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12
0
0
0
0
Sheet1
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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1
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2
3
2
Sheet1
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
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8
1
1
0
Sheet1
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
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2
3
6
1
0
Sheet1
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
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2
1
3
2
0
Sheet1
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| D | |
| E |
| A | 12 | ||
| B | 0 | ||
| C | 0 | ||
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| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 0 | ||
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| E | 13 |
| A | |
| B | |
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| D | |
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| A | 13 | ||
| B | 0 | ||
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| A | |
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| D | |
| E |
| A | 1 | ||
| B | 0 | ||
| C | 14 | ||
| D | 0 | ||
| E | 0 |
| A | |
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| C | |
| D | |
| E |
Chart1
1
0
14
0
0
Sheet1
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis37 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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1
2
2
3
2
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
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1
1
0
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
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2
3
6
1
0
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| A | 12 | ||
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| E | 13 |
| A | |
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| A | 13 | ||
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| A | 1 | ||
| B | 0 | ||
| C | 14 | ||
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| D | |
| E |
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
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0 0
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2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
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0
0
2
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DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
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2
0
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
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0
0
1
4
1
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DTU Compute
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
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3
0
3
0
0
Sheet1
DTU Compute
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
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DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| B | 8 | ||
| C | 1 | ||
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| E |
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| C | 2 | ||
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| E |
| A | 0 | ||
| B | 13 | ||
| C | 0 | ||
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| A | 12 | ||
| B | 0 | ||
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| D | 0 | ||
| E | 13 |
| A | |
| B | |
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| D | |
| E |
| A | 13 | ||
| B | 0 | ||
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| E |
| A | 1 | ||
| B | 0 | ||
| C | 14 | ||
| D | 0 | ||
| E | 0 |
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Correlation on images ndash no normalisation 2A) 010B) 3 3C) 6 2D) 10 15E) 11 32
13
0 0 0 0
A B C D E
-1 -2 -10 0 01 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Chart1
13
0
0
0
0
Sheet1
DTU Compute
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Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
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0
0
0
0
13
Sheet1
DTU Compute
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
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12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
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2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
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2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
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2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
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2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
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2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
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2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
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DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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2122018Introduction to Medical Image Analysis37 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
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2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
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0
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0
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2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
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2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
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2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
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2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
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2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
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0 0
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2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
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0
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0
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2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
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0
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2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
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1
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0
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DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
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DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
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| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
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| B | |
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| D | |
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| A | 0 | ||
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| B | 8 | ||
| C | 1 | ||
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| C | 0 | ||
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| B | 0 | ||
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| A | 0 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 13 |
| A | |
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| A | 13 | ||
| B | 0 | ||
| C | 0 | ||
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| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis37 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
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CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
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0
0
0
0
13
Sheet1
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Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
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Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
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12
0
0
0
0
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Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
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Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
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2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
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2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
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2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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2
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2
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Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
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Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
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Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
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Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
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02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
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02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
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0
0
1
0
0
Sheet1
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02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
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0
0
2
6
0
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02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
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2
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2
0
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02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
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0
0
1
4
1
Sheet1
DTU Compute
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02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
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3
0
3
0
0
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Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
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0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| A | 0 | ||
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| A | |
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| E |
| A | 2 | ||
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| C | 6 | ||
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| E |
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| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
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| E |
| A | 1 | ||
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| E |
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| B | 13 | ||
| C | 0 | ||
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| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 12 | ||
| B | 0 | ||
| C | 0 | ||
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| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 13 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 13 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
DTU Compute
2122018Introduction to Medical Image Analysis37 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
h
f
Correlation operator
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
119892119892 119909119909119910119910 = 119891119891 119909119909119910119910 ∘ ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
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0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
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2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| B | 1 | ||
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| B | |
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| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
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| C | 1 | ||
| D | 0 | ||
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| E |
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| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
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| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 13 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 12 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 13 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis38 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
fh(00)
f(21)
Example g(21)
i = -1 j = -1
i = 1 j = 0
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
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0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| C | 1 | ||
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| E | 13 |
| A | |
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| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis39 DTU Compute Technical University of Denmark
Mathematics of 2D Correlation
1 2 11 3 11 2 1
0 2 4
1 2 2
1 6 3
1 2 1
2 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
h
f
119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892(119909119909119910119910 = 1 2 + 2 0 + 1 1 + 1 1 + 3 4 + 1 2 + 1 0 + 2 1 + 1 0
DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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DTU Compute
2122018Introduction to Medical Image Analysis40 DTU Compute Technical University of Denmark
119892119892 119909119909 119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
119892119892 22 =ℎ minus1minus1 sdot 119891119891 11 + ℎ 0minus1 sdot 119891119891 21 + ℎ 1minus1 sdot 119891119891 31 +ℎ minus10 sdot 119891119891 12 + ℎ 00 sdot 119891119891 22 + ℎ 10 sdot 119891119891 32 +ℎ minus11 sdot 119891119891 13 + ℎ 01 sdot 119891119891 23 + ℎ 11 sdot 119891119891 33
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
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2
Sheet1
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2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
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0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
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1
0
Sheet1
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2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
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2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| B | 8 | ||
| C | 1 | ||
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| A | |
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| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
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| D | |
| E |
| A | 0 | ||
| B | 13 | ||
| C | 0 | ||
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| E | 0 |
| A | |
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| C | |
| D | |
| E |
| A | 12 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 13 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis41 DTU Compute Technical University of Denmark
2D Kernel Normalisation
1 2 11 3 11 2 1
h
Normalisation factor
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
119909119909
119910119910
ℎ(119909119909119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| C | 1 | ||
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| E | 13 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis42 DTU Compute Technical University of Denmark
CorrelationA) 50122B) 123001C) 11233D) 2550E) 90454
0 0 0 0
13
A B C D E
Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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Chart1
0
0
0
0
13
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
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0
8
1
1
0
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DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
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2
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6
1
0
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DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
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0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
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2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
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0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| C | 1 | ||
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| A | 12 | ||
| B | 0 | ||
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| A | |
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| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 13 |
| A | |
| B | |
| C | |
| D | |
| E |
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
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0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| C | 0 | ||
| D | 0 | ||
| E | 13 |
DTU Compute
2122018Introduction to Medical Image Analysis44 DTU Compute Technical University of Denmark
Smoothing filters
Also know as ndash Smoothing kernel Mean filter Low pass filter blurring
The simplest filter ndash Spatial low pass filterndash Removes high frequencies
Another mask ndash Gaussian filter
1 1 1
1 1 1
1 1 191
1 2 1
2 4 2
1 2 1161
Why Gaussian
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
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1
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2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| E |
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| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 13 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 12 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis45 DTU Compute Technical University of Denmark
Use of smoothing
3x3 7x7 11x11 15x15
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
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| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
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| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 13 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 12 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis46 DTU Compute Technical University of Denmark
Use of smoothing Large kernels smooth more Removes high frequency information Good at enhancing big structures
3x3 15x15
DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
Chart1
12
0
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
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DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
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DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
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DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
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DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
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0
6
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DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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DTU Compute
2122018Introduction to Medical Image Analysis47 DTU Compute Technical University of Denmark
Mean filterA) 166B) 113C) 12D) 51E) 245
0 0 0 0
A B C D E
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0
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DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
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2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
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DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
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2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
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0
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0
0
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DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
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2
1
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2
0
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DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| E |
Chart1
12
0
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DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| A | |
| B | |
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| D | |
| E |
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
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| E | 6 |
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| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 13 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 12 | ||
| B | 0 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
DTU Compute
2122018Introduction to Medical Image Analysis49 DTU Compute Technical University of Denmark
Border handling
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input Output
No values at the border
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
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| A | 3 | ||
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| B | |
| C | |
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| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
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| E | 1 |
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| B | |
| C | |
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| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 13 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis50 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
0 0 0 0 000000
bull Zero padding ndash what happens
bull Zero is black ndash creates dark border around the image
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| B | 0 | ||
| C | 5 | ||
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| E | 6 |
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| C | 2 | ||
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| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
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| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
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| E | 0 |
| A | |
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| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
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| E |
| A | 0 | ||
| B | 13 | ||
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| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis51 DTU Compute Technical University of Denmark
Correlation with zero paddingA) 5 8B) 12 16C) 13 3D) 15 21E) 17 12
1 2 11 3 11 2 1
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
0
13
0 0 0
A B C D E
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
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| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 13 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
Chart1
0
13
0
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 13 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 13 | ||
| C | 0 | ||
| D | 0 | ||
| E | 0 |
DTU Compute
2122018Introduction to Medical Image Analysis53 DTU Compute Technical University of Denmark
Border handling ndash extend the input
0 2 41 2 2
1 6 3
1 2 12 5 3
2 1 3
1 3 1
2 2 2
0 1 3
1 2 0
2 1 4
1 0 1
Input
1 2 0 1 311211
bull Reflection
bull Normally better than zero padding
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
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| E |
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| C | 3 | ||
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| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis54 DTU Compute Technical University of Denmark
Template Matching Template
ndash Skabelon paring dansk Locates objects in images
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis55 DTU Compute Technical University of Denmark
Template Matching The correlation between the template and the input
image is computed for each pixel
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
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| E | 6 |
| A | |
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| E |
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| C | 1 | ||
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| E | 1 |
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| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
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| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis56 DTU Compute Technical University of Denmark
Template Matching The pixel with the highest value is found in the
output imagendash Here is the highest correlation
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
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| B | 0 | ||
| C | 1 | ||
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| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
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| A | |
| B | |
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| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis57 DTU Compute Technical University of Denmark
Template Matching This corresponds to the found pattern in the input
image
Input
Template
Output Correlation image
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
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| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis58 DTU Compute Technical University of Denmark
Problematic Correlation Correlation matching has problem with light areas ndash
why
Input (f)
Template (h)
Output Correlation imageFake max
Real max
Very High119892119892 119909119909119910119910 = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) 119891119891(119909119909 + 119894119894119910119910 + 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis59 DTU Compute Technical University of Denmark
Normalised Cross Correlation
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
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| B | 0 | ||
| C | 3 | ||
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| E | 0 |
| A | |
| B | |
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| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis60 DTU Compute Technical University of Denmark
Length of template Vector length
ndash Put all pixel values into a vectorndash Compute the length of this vector
Describes the intensity of the templatendash Bright template has a large lengthndash Dark template has a small length
Template (h)
Length of template = 119895119895=minus119877119877
119877119877
119894119894=minus119877119877
119877119877
ℎ(119894119894 119895119895) ℎ(119894119894 119895119895)
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis61 DTU Compute Technical University of Denmark
Length of image patch Vector length based on pixel values in image patch Describes the intensity of the image patch
Template (h)
Input (f) with patch
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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| B | 8 | ||
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| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis62 DTU Compute Technical University of Denmark
Normalised Cross Correlation The length of the image patch and the length of
template normalise the NCC If the image is very bright the NCC will be ldquopulled
downrdquo
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
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| E |
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| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis63 DTU Compute Technical University of Denmark
Normalised Cross Correlation NCC will be between
ndash 0 No similarity between template and image patchndash 1 Template and image patch are identical
Input (f)
Template (h)
Output Correlation image
Real max
NCC x y =Correlation
Length of image patch sdot Length of template
DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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DTU Compute
2122018Introduction to Medical Image Analysis64 DTU Compute Technical University of Denmark
Normalised Cross Correlation
A) 010B) 033C) 083D) 062E) 098
Der udfoslashres template matching med og normalized crosscorrelation (NCC) beregnes Hvad bliver NNC i den markerede pixel
1
2 2
3
2
A B C D E
Chart1
1
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2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
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| E |
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| E |
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| A | |
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| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
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| E |
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| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
| A | |
| B | |
| C | |
| D | |
| E |
Chart1
1
2
2
3
2
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
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| E |
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| B | 0 | ||
| C | 3 | ||
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| B | |
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| D | |
| E |
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| C | 1 | ||
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| E | 1 |
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| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
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| E |
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| E |
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| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
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| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
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| C | |
| D | |
| E |
| A | 1 | ||
| B | 2 | ||
| C | 2 | ||
| D | 3 | ||
| E | 2 |
DTU Compute
2122018Introduction to Medical Image Analysis66 DTU Compute Technical University of Denmark
Edges An edge is where there is a
high change in gray level values
Objects are often separated from the background by edgesGray level profile
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis67 DTU Compute Technical University of Denmark
Edges The profile as a function f(d) What value is high when there
is an edgendash The slope of fndash The slope of the tangent at d
f(d)
119891119891prime(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
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| A | |
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| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis68 DTU Compute Technical University of Denmark
Finite Difference Definition of slope
Approximation
Simpler approximationh = 1
119891119891prime 119889119889 = limℎrarr0
119891119891 119889119889 + ℎ minus 119891119891(119889119889)ℎ
119891119891prime 119889119889 asymp119891119891 119889119889 + ℎ minus 119891119891(119889119889)
ℎ
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
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| E |
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| E | 1 |
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| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
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| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis69 DTU Compute Technical University of Denmark
Edges Discrete approximation of frsquo(d) Can be implemented as a filter
-1 0 1
f(d-2) f(d-1) f(d) f(d+1) f(d+2) f(d+3)
f(d)
119891119891prime 119889119889 asymp 119891119891 119889119889 + 1 minus 119891119891(119889119889)
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
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| C | 3 | ||
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| C | 1 | ||
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| E | 1 |
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| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
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| E | 0 |
| A | |
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| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis70 DTU Compute Technical University of Denmark
Edges in 2D Changes in gray level values
ndash Image gradientndash Gradient is the 2D derivative of a 2D function f(xy)ndash Equal to the slope of the imagendash A steep slope is equal to an edge
120571120571119891119891 119909119909119910119910 = 119866(119892119892119909119909119892119892119910119910)
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
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| E | 6 |
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| E |
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| B | 0 | ||
| C | 1 | ||
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| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
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| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis71 DTU Compute Technical University of Denmark
Edge filter kernel The Prewitt filter is a typical
edge filter Output image has high
values where there are edges
-1 0 1
-1 0 1
-1 0 1Vertical Prewitt filter
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
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| E | 0 |
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| B | |
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| E |
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| E |
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| B | 1 | ||
| C | 3 | ||
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| B | |
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| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis72 DTU Compute Technical University of Denmark
Prewitt filter
Original Prewitt Prewitt
Hot colormapSmooth 15x15
Smooth 15x15
Prewitt
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
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| B | 0 | ||
| C | 3 | ||
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| E | 0 |
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| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
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| E | 0 |
| A | |
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| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
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| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis73 DTU Compute Technical University of Denmark
Edge detection Edge filter
ndash Prewitt for example Thresholding
ndash Separate edges from non-edges
Output is binary imagendash Edges are white
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
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| E | 6 |
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| B | |
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| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
DTU Compute
2122018Introduction to Medical Image Analysis74 DTU Compute Technical University of Denmark
Edge filtering
A) MeanB) Horizontal SobelC) Vertical PrewittD) Horisontal PrewittE) Median
Der udfoslashres en filtrering (correlation) med et filter paring billedet Der bruges ikke coefficient normalization Den filtrerede vaeligrdi i den markerede pixel bliver -119 Hvilket et 3x3 filter blev benyttet
0
8
1 10
A B C D E
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
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| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
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| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
Chart1
0
8
1
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
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| B | 1 | ||
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| E | 0 |
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| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
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| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
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| E | 0 |
| A | |
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| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
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| E |
| A | 0 | ||
| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
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| B | 8 | ||
| C | 1 | ||
| D | 1 | ||
| E | 0 |
DTU Compute
2122018Introduction to Medical Image Analysis76 DTU Compute Technical University of Denmark
02511 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
23
6
10
A B C D E
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
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| C | 3 | ||
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| B | |
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| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
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| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
Chart1
2
3
6
1
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
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| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
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| C | 2 | ||
| D | 6 | ||
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| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
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| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 3 | ||
| C | 6 | ||
| D | 1 | ||
| E | 0 |
DTU Compute
2122018Introduction to Medical Image Analysis77 DTU Compute Technical University of Denmark
02512 - Niveau af dagens emne forelaeligsingA) Meget letB) LetC) PassendeD) SvaeligrtE) Meget svaeligrt
0 0
1
0 0
A B C D E
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
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DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
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DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
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| B | |
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| E |
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| C | 3 | ||
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| E | 0 |
| A | |
| B | |
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| D | |
| E |
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| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
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| B | |
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| E |
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| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
Chart1
0
0
1
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
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1
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2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
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DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 0 | ||
| E | 0 |
DTU Compute
2122018Introduction to Medical Image Analysis78 DTU Compute Technical University of Denmark
02511 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
0 0
2
6
0
A B C D E
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
Chart1
0
0
2
6
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 2 | ||
| D | 6 | ||
| E | 0 |
DTU Compute
2122018Introduction to Medical Image Analysis79 DTU Compute Technical University of Denmark
02512 - Mit eget udbytte af dagenA) Jeg har laeligrt meget lidtB) Lidt laeligringC) PassendeD) God laeligringE) Jeg har laeligrt meget
2
1
3
2
0
A B C D E
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
Chart1
2
1
3
2
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 2 | ||
| B | 1 | ||
| C | 3 | ||
| D | 2 | ||
| E | 0 |
DTU Compute
2122018Introduction to Medical Image Analysis80 DTU Compute Technical University of Denmark
02511 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
0 0
1
4
1
A B C D E
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
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| A | |
| B | |
| C | |
| D | |
| E |
Chart1
0
0
1
4
1
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
| A | |
| B | |
| C | |
| D | |
| E |
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 0 | ||
| B | 0 | ||
| C | 1 | ||
| D | 4 | ||
| E | 1 |
DTU Compute
2122018Introduction to Medical Image Analysis81 DTU Compute Technical University of Denmark
02512 - Sidste uges oslashvelser (pixelwiseoperations)A) Meget letteB) LetteC) PassendeD) SvaeligreE) Meget svaeligre
3
0
3
0 0
A B C D E
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
Chart1
3
0
3
0
0
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
| A | |
| B | |
| C | |
| D | |
| E |
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
| A | 3 | ||
| B | 0 | ||
| C | 3 | ||
| D | 0 | ||
| E | 0 |
DTU Compute
2122018Introduction to Medical Image Analysis82 DTU Compute Technical University of Denmark
Denne uges brug af clickersA) click click click ndash jeg bliver
sindssygB) For meget clickC) PassendeD) Jeg vil gerne have lidt mereE) Jeg kan ikrsquo faring nok
0 0
5
0
6
A B C D E
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
Chart1
0
0
5
0
6
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
| A | |
| B | |
| C | |
| D | |
| E |
Sheet1
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
| A | 0 | ||
| B | 0 | ||
| C | 5 | ||
| D | 0 | ||
| E | 6 |
DTU Compute
2122018Introduction to Medical Image Analysis83 DTU Compute Technical University of Denmark
What can you do after today Describe the difference between point processing and neighbourhood
processing Compute a rank filtered image using the min max median and
difference rank filters Compute a mean filtered image Decide if median or average filtering should be used for noise removal Choose the appropriate image border handling based on a given input
image Implement and apply template matching Compute the normalised cross correlation and explain why it should be
used Apply given image filter kernels to images Use edge filters on images Describe finite difference approximation of image gradients Describe the concept of edge detection
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
DTU Compute
2122018Introduction to Medical Image Analysis84 DTU Compute Technical University of Denmark
Oslashvelsesrapport over oslashvelse 4 - 02511 Oslashvelser taeligller cirka 30 af karakteren Skal afleveres senest den 7 marts kl 2359 Kan afleveres af 1 eller 2 personer
ndash I skal skrive jeres egen oslashvelsesrapportndash Brug mindst et af jeres egne billeder
Afleveres som PDF Lad vaeligre med at kopiere opgaveteksten ind i jeres
opgave Put Matlab kode i et Appendix og henvis fra teksten Eksempel rapport over oslashvelse findes paring
hjemmesiden under eksamen
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
DTU Compute
2122018Introduction to Medical Image Analysis85 DTU Compute Technical University of Denmark
Exercises
- Introduction to Medical Image Analysis
- Lecture 4 ndash Neighbourhood Processing
- What can you do after today
- Point processing
- Point processing
- Neighbourhood processing
- Use of filtering
- Salt and pepper noise
- Salt and pepper noise
- What is so special about noise
- Noise ndash go away
- What is the median Value
- The median value
- Noise away ndash the median filter
- Noise removal ndash average filter
- Noise removal ndash median filter
- Image Filtering
- Median filter
- Rank filters
- Median filter
- Correlation
- Why do we need similarity
- Correlation (1D)
- Correlation (1D)
- Normalisation
- Normalisation
- Correlation on images
- Correlation on images
- Correlation on images
- Draw your filter
- Correlation on images ndash no normalisation
- Correlation on images ndash no normalisation 2
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Mathematics of 2D Correlation
- Slide Number 40
- 2D Kernel Normalisation
- Correlation
- Smoothing filters
- Use of smoothing
- Use of smoothing
- Mean filter
- Border handling
- Border handling ndash extend the input
- Correlation with zero padding
- Border handling ndash extend the input
- Template Matching
- Template Matching
- Template Matching
- Template Matching
- Problematic Correlation
- Normalised Cross Correlation
- Length of template
- Length of image patch
- Normalised Cross Correlation
- Normalised Cross Correlation
- Normalised Cross Correlation
- Edges
- Edges
- Finite Difference
- Edges
- Edges in 2D
- Edge filter kernel
- Prewitt filter
- Edge detection
- Edge filtering
- 02511 - Niveau af dagens emne forelaeligsing
- 02512 - Niveau af dagens emne forelaeligsing
- 02511 - Mit eget udbytte af dagen
- 02512 - Mit eget udbytte af dagen
- 02511 - Sidste uges oslashvelser (pixelwise operations)
- 02512 - Sidste uges oslashvelser (pixelwise operations)
- Denne uges brug af clickers
- What can you do after today
- Oslashvelsesrapport over oslashvelse 4 - 02511
- Exercises
-
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