image_enhancement_spatial

CIS 601

Image ENHANCEMENTin the

SPATIAL DOMAIN

Dr. Rolf Lakaemper

Most of these slides base on the book

Digital Image Processingby Gonzales/Woods

Chapter 3

Introduction

Image Enhancement ?

• enhance otherwise hidden information• Filter important image features• Discard unimportant image features

Spatial Domain ?

• Refers to the image plane (the ‘natural’ image)

• Direct image manipulation

Remember ?

A 2D grayvalue - image is a 2D -> 1D function,

v = f(x,y)

Remember ?

As we have a function, we can apply operators to this function, e.g.

T(f(x,y)) = f(x,y) / 2

Operator Image (= function !)

Remember ?

T transforms the given image f(x,y)into another image g(x,y)

f(x,y) g(x,y)

Spatial Domain

The operator T can be defined over

• The set of pixels (x,y) of the image• The set of ‘neighborhoods’ N(x,y) of

each pixel• A set of images f1,f2,f3,…

Operation on the set of image-pixels

6 8 2 0

12 200 20 10

3 4 1 0

6 100 10 5

Spatial Domain

(Operator: Div. by 2)

Operation on the set of ‘neighborhoods’ N(x,y) of each pixel

6 8 2 0

12 200 20 10

226

Spatial Domain

6 8

12 200

(Operator: sum)

Operation on a set of images f1,f2,…

6 8 2 0

12 200 20 10

Spatial Domain

5 5 1 0

2 20 3 4

11 13 3 0

14 220 23 14

(Operator: sum)

Operation on the set of image-pixels

Remark: these operations can also be seen as operations on the neighborhood of a pixel (x,y), by defining the neighborhood as the pixel itself.

• The simplest case of operators• g(x,y) = T(f(x,y)) depends only on the value

of f at (x,y)• T is called a

gray-level or intensity transformation function

Spatial Domain

Basic Gray Level Transformations

• Image Negatives• Log Transformations• Power Law Transformations• Piecewise-Linear Transformation

Functions

For the following slides L denotes the max. possible gray value of the image, i.e. f(x,y) [0,L]

Transformations

Image Negatives: T(f)= L-f

Transformations

Input gray level

Out

put g

ray

leve

l T(f)=L-f

Log Transformations:T(f) = c * log (1+ f)

Transformations

Log Transformations

Transformations

InvLog Log

Log Transformations

Transformations

Power Law TransformationsT(f) = c*f

Transformations

• varying gamma () obtains family of possible transformation curves

> 0 • Compresses dark values• Expands bright values

< 0 • Expands dark values• Compresses bright values

Transformations

• Used for gamma-correction

Transformations

• Used for general purpose contrast manipulation

Transformations

Piecewise Linear Transformations

Transformations

Thresholding Function

g(x,y) = L if f(x,y) > t,0 else

t = ‘threshold level’


Input gray level

Out

put g

ray

leve

l

Gray Level Slicing

Purpose: Highlight a specific range of grayvalues

Two approaches:

1. Display high value for range of interest, low value else (‘discard background’)

2. Display high value for range of interest, original value else (‘preserve background’)


Gray Level Slicing


Bitplane Slicing

Extracts the information of a single bitplane



BP 7

BP 5

BP 0

Exercise:

• How does the transformation function look for bitplanes 0,1,… ?

• What is the easiest way to filter a single bitplane (e.g. in MATLAB) ?


Histograms

Histogram Processing

gray level

Num

ber

of P

ixel

s1 4 5 0

3 1 5 1

Histograms

Histogram Equalization:

• Preprocessing technique to enhance contrast in ‘natural’ images

• Target: find gray level transformation function T to transform image f such that the histogram of T(f) is ‘equalized’

Histogram Equalization

Equalized Histogram:

The image consists of an equal number of pixels for every gray-

value, the histogram is constant !


Example:

We are looking forthis transformation !

T


Target: Find a transformation T to transform the

grayvalues g1[0..1] of an image I to grayvalues g2 = T(g1) such that the histogram is equalized, i.e. there’s an equal amount of pixels for each grayvalue.

Observation (continous model !):Assumption: Total image area = 1 (normalized). Then:

The area(!) of pixels of the transformed image in the gray-value range 0..g2 equals the gray-value g2.


The area(!) of pixels of the transformed image in the gray-value range 0..g2 equals the gray-value g2.

Every g1 is transformed to a grayvalue that equals the area (discrete: number of pixels) in the image covered by pixels having gray-values from 0 to g1.

The transformation T function t is the

area- integral: T: g2 = 0..g1 I da


Discrete:

g1 is mapped to the (normalized) number of pixels having grayvalues 0..g1 .


Mathematically the transformation is deducted by theorems in

continous (not discrete) spaces.

The results achieved do NOT hold for discrete spaces !

(Why ?)

However, it’s visually close.


Conclusion:

• The transformation function that yields an image having an equalized histogram is the integral of the histogram of the source-image

• The discrete integral is given by the cumulative sum, MATLAB function: cumsum()

• The function transforms an image into an image, NOT a histogram into a histogram ! The histogram is just a control tool !

• In general the transformation does not create an image with an equalized histogram in the discrete case !

Operations on a set of images

Operation on a set of images f1,f2,…

6 8 2 0

12 200 20 10

5 5 1 0

2 20 3 4

11 13 3 0

14 220 23 14

(Operator: sum)


Logic (Bitwise) Operations

ANDOR

NOT


The operators AND,OR,NOT are functionally complete:

Any logic operator can be implemented using only these 3 operators


Any logic operator can be implemented using only these 3 operators:

A B Op

0 0 1

0 1 1

1 0 0

1 1 0

Op=

NOT(A) AND NOT(B)

OR

NOT(A) AND B


Image 1 AND Image 2

1 2 3 9

7 3 6 4

1 1 1 1

2 2 2 2

1 0 1 1

2 2 2 0

(Operator: AND)


Image 1 AND Image 2:Used for Bitplane-Slicing and

Masking


Exercise: Define the mask-image, that transforms image1 into image2 using

the OR operand

1 2 3 9

7 3 6 4

255 2 7 255

255 3 7 255

(Operator: OR)

Operations

Arithmetic Operations on a set of images

1 2 3 9

7 3 6 4

1 1 1 1

2 2 2 2

2 3 4 10

9 5 8 6

(Operator: +)

Operations

Exercise:

What could the operators + and – be used for ?

Operations

(MATLAB) Example: Operator –

Foreground-Extraction

Operations

(MATLAB)Example: Operator +

Image Averaging

Part 2

CIS 601

Image ENHANCEMENTin the

SPATIAL DOMAIN

Histograms

So far (part 1) :

• Histogram definition• Histogram equalization

Now:

• Histogram statistics

Histograms

Remember:The histogram shows the number of

pixels having a certain gray-value

nu

mbe

r of

pix

els

grayvalue (0..1)

Histograms

The NORMALIZED histogram is the histogram divided by the total number of pixels in the source image.

The sum of all values in the normalized histogram is 1.

The value given by the normalized histogram for a certain gray value can be read as the probability of randomly picking a pixel having that gray value

Histograms

What can the (normalized) histogram tell about the

image ?

Histograms

1.The MEAN VALUE (or average gray level)

M = g g h(g)

1*0.3+2*0.1+3*0.2+4*0.1+5*0.2+6*0.1=2.6

0.30.20.10.0

1 2 3 4 5 6

Histograms

The MEAN value is the average gray

value of the image, the ‘overall brightness appearance’.

Histograms

2. The VARIANCE

V = g (g-M)2 h(g)

(with M = mean)or similar:

The STANDARD DEVIATION

D = sqrt(V)

Histograms

VARIANCE gives a measure about the distribution of the histogram values around the mean.

0.30.20.10.0

0.30.20.10.0

V1 > V2

Histograms

The STANDARD DEVIATION is a value on the gray level axis, showing the average distance of all pixels to the mean

0.30.20.10.0

0.30.20.10.0

D1 > D2

Histograms

VARIANCE and STANDARD DEVIATION of the histogram tell us about the average contrast of the image !

The higher the VARIANCE (=the higher the STANDARD DEVIATION), the

higher the image’s contrast !

Histograms

Example:

Image and blurred version

Histograms

Histograms with MEAN and STANDARD DEVIATION

M=0.73 D=0.32 M=0.71 D=0.27

Histograms

Exercise:

Design an autofocus system for a digital camera !

The system should analyse an area in the middle of the picture and automatically adjust the lens such that this area is sharp.

Histograms

In between the basics…

…histograms can give us a first hint how to create image databases:

Feature Based Coding


• Determine a feature-vector for a given image

• Compare images by their feature-vectors

Two operations need to be defined: a mapping of shape into the feature space and a similarity of feature vectors.

Where are the histograms ?

Representation Feature Extraction Vector Comparison



• Determine a feature-vector for a given image

• Compare images by their feature-vectors

Two operations need to be defined: a mapping of shape into the feature space and a similarity of feature vectors.

HERE !

Question: how can we compare histograms (vectors) ?

Representation HISTOGRAM Histogram Comp.

Vector Comparison

Vector Comparison

,

Vector Comparison

Vector Comparison

Vector Comparison

What’s the meaning of the Cosine Distance with respect to histograms ?

i.e.: what’s the consequence of eliminating the vector’s length information ?

Vector Comparison

More Vector Distances:

• Quadratic Form Distance

• Earth Movers Distance

• Proportional Transportation Distance

• …

Vector Comparison

Histogram Intersection

(non symmetric):

d(h1,h2) = 1 - min(h1 ,h2 ) / h1

Ex.: What could be a huge drawback of image comparison using histogram intersection ?

i i i ii

Histograms

Exercise:

Outline an image database system, using statistical (histogram)

information

Histograms

Discussion:

Which problems could occur if the database consists of the following

images ?

Histograms

Spatial Filtering

End of histograms.

And now to something completely

different …

image_enhancement_spatial

Education

operator image

image gx

transformed image

image enhancementin

gray value

given image fx

total image area

histogram of tf