4.intensity transformations

Digital Image ProcessingIII. Intensity transformations

Hamid LagaInstitut Telecom / Telecom Lille1

hamid@img.cs.titech.ac.jp

http://www.img.cs.titech.ac.jp/~hamid/

In the previous lecture

• Colors• The results of the interaction of light with the surface as observed by a

sensor from a viewing point(the surface absorbs some light, depending on the material properties)

• Image• a 2D function which encodes the color at each location of the image plane

• Digital image• a 2D matrix, each entry is called a pixel, it can have a single value

(grayscale) or a vector of 3 values (color)

• Color spaces• There are many color spaces, from now we will use the RGB color space.

• If we need to use another color space I will mention it explicitly

2

In the this lecture

• Basic processing operations that can be done on an image•

3

Intensity transformations

• Outline

– Basic intensity transformations(Image negatives, log transformations, power-law or Gamma transformations)

– Image histogram(Definitions, histogram equalization, local histogram processing, histogram statistics for image enhancement)

– Your first TP (to be done in Matlab)(Introduction to Matlab, Image manipulation with Matlab, …)

– Summary

4

Some definitions

• Spatial domain• The image plane itself

• Processing in the spatial domain deals with direct manipulation of the image pixels

• Other domains• Frequency domain

• Consider the image as a 2D signal and apply Fourier transform

• Process the image in the transformed domain

• Apply the inverse transform to return into the spatial domain.

• Types of transformations that can be applied to an image• Intensity transformations (this lecture)

• Operate on single pixels

• Spatial filtering (next lecture)• Working on a neighborhood of every pixel

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Basics

• Intensity transformation and spatial filtering

• f : input image

• g: output image

• T: an operator applied to the image f

• In the discrete case• f and g are 2D matrices

of size WxH

• T is a matrix of size wxhwhich is much smaller than WxH

• The value of each pixel of g is the resultof applying the operator T at the correspondinglocation in f.

• If wxh = 1x1 intensity transform

• Otherwise, it’s spatial filtering.

6

),(),( yxfTyxg

Basic intensity transforms

• Definition

• g(x, y) = T (f(x, y))

• Examples

• Image negatives

• Log transformations

• Power-law (Gamma) transformations

• In the following

• Each image is represented with a 2D matrix of L intensity levels. That is each

pixel takes a value between 0 and L-1

7

Basic intensity transforms

• Image negatives

• Reversing the intensity levels

g(x, y) = L – 1 – f(x, y)

– It produces the equivalent of a photographic negatives

8

Input image Its negative

Basic intensity transforms

• Image negatives

• Can be used to enhance white or gray details embedded in dark regions

especially when the black areas are dominant in size

9

Input image Its negative

Basic intensity transforms

• Log transformations

g(x, y) = c . Log (1 + f(x, y))

– c is a constant and f(x, y) >= 0.

10

Input image Its log transform c=1

Basic intensity transforms

• Log transformations

– Maps a narrow range of low intensity values into a wider range of output

values (spreading)

– The opposite is true for the higher values of input levels (compressing)

11 x

Log(

1 +

x)

Basic intensity transforms

• Log transformations

– Maps a narrow range of low intensity values into a wider range of output

values (spreading)

– The opposite is true for the higher values of input levels (compressing)

12

Input imagePixel values range from 0 to 106

Its log transform with c=1Values range from 0 to 6.2

Basic intensity transforms

• Power-law (Gamma) transformations

g(x, y) = c . f(x, y)

– c and are positive constants.

13

Basic intensity transforms

• Power-law (Gamma) transformations

g(x, y) = c . f(x, y)

– c and are positive constants.

14

Original image and results of applyingGamma transformations with

c=1 and Gamma = 0.6, 0.4, and 0.3(increasing the dynamic range of

low intensities)

Basic intensity transforms

• Power-law (Gamma) transformations

g(x, y) = c . f(x, y)

– c and are positive constants.

15

Original image and results of applyingGamma transformations with c=1 and Gamma = 3, 4, and 5

(increasing the dynamic range of highintensities)

Basic intensity transforms

• Piecewise linear transformation functions

– Piecewise linear functions can be arbitrary complex

more flexibility in the design of the transformation

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Piecewise linear transformations

• Contrast stretching

– Low contrast images can result from poor illumination

17

Low contrast imageReduce

intensity

ReduceintensityIncrease

intensity

Piecewise linear transformations

• Contrast stretching

– Low contrast images can result from poor illumination

18

Low contrast image Contrast stretching

Intensity transformations

• Outline

– Basic intensity transformations(Image negatives, log transformations, power-law or Gamma transformations, piecewise-linear transformation functions)

– Image histogram(Definitions, histogram equalization, histogram matching)

– Your first TP (to be done in Matlab)(Introduction to Matlab, Image manipulation with Matlab, …)

– Summary

19

Image histogram - definitions

• Histogram– The histogram function is defined over all possible

intensity levels. – For each intensity level, its value is equal to the number of

the pixels with that intensity.

• Applications

– Image enhancement, compression, segmentation

– Very popular tool for realtime image processing

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Image histogram - examples

• Consider the following 5x5 image

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1 8 4 3 4

1 1 1 7 8

8 8 3 3 1

2 2 1 5 2

1 1 8 5 2

Image histogram - example

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1 8 4 3 4

1 1 1 7 8

8 8 3 3 1

2 2 1 5 2

1 1 8 5 2

1 2 3 4 5 6 7 8

Image histogram - examples

23

Graph of the histogram function

Original image

Image histogram - examples

24

Graph of the histogram function

Original image

Image histogram - examples

25

Graph of the histogram function

Original image

Image histogram - examples

26

Graph of the histogram function

Original image

Image histogram - examples

27

Uniform distribution

Image histogram - definitions

• Normalized histogram– The normalised histogram function is the histogram function divided

by the total number of the pixels of the image

where n is the number of pixels in the image, and r is an intensity level.

– It is the frequency of appearance of each intensity level (color) in the image

– It gives a measure of how likely is for a pixel to have a certain intensity.

That is, it gives the probability of occurrence the intensity.– The sum of the normalised histogram function over the range of all

intensities is 1.

28

n

rhrp

)()(

Cumulative Distribution Function (CDF)

• If h is the normalized histogram of an image, the CDF of h is

define as:

29

Histogram h CDF of the normalized h

i

j

h ihiCDF0

)()(

Histogram equalization

• Goal• Increase the global contrast of an image by design a transformation (mapping) T

that that makes pixel intensities uniformly distributed over the whole range [0, L-1]

• Input• An image f of intensity levels in [0, L-1]

• Intensity levels can be viewed as random variables

• Let’s hf be the normalized histogram of f. hf is also the probability distribution function which we denote by Pf

• We denote by r intensity values of f

• Output• An image g = T(f) such that the intensities of g are uniformly over the range [0, L-

1].

• This will improve the contrast by spreading out the most frequent intensity values

• We denote by s = T(r).

30

Review of probabilities

• Uniform probability distribution• A probability distribution P is uniform if all events have the same probability of

occurrence.

• If N is the number of events then P(x) = 1 / N for every event x.

31

Intensities

Fre

qu

en

cy o

f

occ

urr

en

ce

0 1

Ideal equalized

histogram function

Histogram equalization

• We want to compute• An image g = T(f) such that the intensities of g are better distributed on the

histogram hg

• hg as a PDF, the intensities of g should follow a uniform distribution.

• Requirements for a good mapping T

• Condition A: T is required to be a monotonically increasing function in the interval [0, L-1]

• That is if x < y T(x) <= T(y)

• Avoids reversing intensities

• Condition B: x [0, L-1] T(x) [0, L-1] • This guarantees that the range of output intensities are the same as the range of input

intensities.

• Condition C: You may require strict monotonicity• x < y T(x) < T(y)

• This guarantees the mapping (or transformation )T is one-to-one.

32

Histogram equalization

• Example of good mapping T

33

Input intensity

Ou

tpu

t in

ten

sity

Input intensity

Ou

tpu

t in

ten

sity

Conditions A, B, and C are satisfied

Conditions A and B are satisfied, but not C

Histogram equalization

• Consider the following transformation

• In the discrete case

– That is, we add the values of the normalised histogram function from 1 to k to find where the intensity will be mapped.

• This is the cumulative distribution function

• Short quiz:• Prove that the transformation T defined as above satisfies the condition A

and condition B.

• Does it satisfy condition C?

34

r

f dwwpLrTs0

)()1()(

r

f rhLrTs0

)()1()(

Histogram equalization

• Solution to the Quiz

35

Histogram equalization

• Interesting property

• If we apply the transformation T to the image f we get an image g of

histogram hg such that

• This is a uniform distribution !!!

36

1

1)(],1,1[

LshLs g

Histogram equalization – Algorithm

• Input

• A discrete image f of L levels of intensity

• Algorithm

• Step1: Compute the normalized histogram hf of input image f

• Step 2: Compute the CDF of hf, denoted CDFhf

• Step 3: Find a transformation T: [0, L-1] [0, L-1] such that

• Step 4: Apply the transformation on each pixel of the input image f

• Propert ies

• The procedure is fully automatic and very fast

• Suitable for real-time applications

37

r

q

hf rCDFLqhLrT0

)()1()()1()(

Histogram equalization – Example

• Do histogram equalization on the 5x5 image with integer intensities in the range between one and eight

38

1 8 4 3 4

1 1 1 7 8

8 8 3 3 1

2 2 1 5 2

1 1 8 5 2

Histogram equalization – Example

• Do histogram equalization on the 5x5 image with integer intensities in the range between one and eight

39

1 8 4 3 4

1 1 1 7 8

8 8 3 3 1

2 2 1 5 2

1 1 8 5 2

20.0)(

04.0)(

00.0)(

08.0)(

08.0)(

12.0)(

16.0)(

32.0)(

8

7

6

5

4

3

2

1

rp

rp

rp

rp

rp

rp

rp

rp

)(

)(

)(

)(

68.008.012.016.032.0)(

60.012.016.032.0)(

48.016.032.0)(

32.0)(

8

7

6

5

4

3

2

1

rT

rT

rT

rT

rT

rT

rT

rT

Intensity transformation functionNormalised

histogram function

Histogram equalization – Example

• Do histogram equalization on the 5x5 image with integer intensities in the range between one and eight

40

20.0)(

04.0)(

00.0)(

08.0)(

08.0)(

12.0)(

16.0)(

32.0)(

8

7

6

5

4

3

2

1

rp

rp

rp

rp

rp

rp

rp

rp

Intensity transformation function

Normalised histogram function

00.1)(

80.0)(

76.0)(

76.0)(

68.0)(

60.0)(

48.0)(

32.0)(

8

7

6

5

4

3

2

1

rT

rT

rT

rT

rT

rT

rT

rT The 32% of the pixels have intensity r1. We expect them to cover 32% of the possible intensities.

The 48% of the pixels have intensity r2 or less. We expect them to cover 48% of the possible intensities.

The 60% of the pixels have intensity r3 or less. We expect them to cover 60% of the possible intensities.

……………………………

Histogram equalization – Example

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Histogram equalization – Example

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Histogram equalization – Example

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Histogram equalization of color images

• The above procedure is for grayscale images

• For color images

• Apply separately the procedure to each of the three channels

• will yield in a dramatic change in the color balance

• NOT RECOMMENED unless you have a good reason for doing that.

• Recommended method

• Convert the image into another color space such as the Lab color space OR

HSL/HSV color space

• Apply the histogram equalization on the Luminance channel

• Convert back into the RGB color space.

44

Lab color space

• Lab color space is designed to approximate the human vision

• The L component matches the human perception of lightness

• The Lab space is much larger than the gamut of computer displays.

• Conversions

• Will be covered in a special lecture on Colors.

45

Histogram matching

• Histogram equalization tries to make the intensity distribution uniform. – Because of the irregular initial distribution, usually this is not possible.– Sometimes histogram equalization introduces artefacts,

(see the example with the photograph of the moon).

46

Intensities

Fre

qu

en

cy o

f

occ

urr

en

ce

0 1

Ideal equalized

histogram function

Histogram matching

• Instead– We can specify the function we want for the histogram

47

Intensities

Fre

qu

en

cy o

f

occ

urr

en

ce

0 1

hf: initial (source)histogram

P: specified (target)histogram

Find a transformation T such that the normalized

histogram of T(f) matches the probability distribution P.

P can be the normalized histogram of another image g,

then T transfers the histogram of g to f.

Histogram matching

• Goal

• Given an image f with a normalized histogram hf

• Given a probability distribution P

• Find a transformation T such that the normalized histogram of T(f)

matches the probability distribution P.

• Remarks

• If P is a uniform distribution then T is the same as the histogram

equalization.

48

Histogram matching - procedure

• Input

• A source image f and a target distribution function P

• P can be the normalized histogram of another image g

• Procedure

• Calculate the histogram hf of f

• Calculate the CDFf of f and the CDFP of P

• For each gray level x [0, L-1], find the gray level y for which

• CDFf(x) = CDFP(y)

• We denote this mapping by M

• Can be implemented as a lookup table

• Apply the mapping M to all pixels of the input image f

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Histogram matching - examples

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Histogram matching - examples

• Undoing with histogram equalization

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Summary

• What we have seen

• Basic intensity transformations applied to individual pixels

• Histogram

• A very powerful tool in image processing

• A connection to probability and statistics

• Histogram equalization

• Histogram matching

• Enables the matching of the histogram of one image to the histogram of another

image

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Next

• Spatial filtering, convolution, and enhancement

– Fundamentals(Kernel, convolution, filtering)

– Smoothing (Linear / non-linear filters)

– Sharpening (The Laplacian - 2nd order derivatives, the gradient - first order derivatives)

– Image pyramids (The Gaussian pyramid)

– Filtering in the frequency domain (just a short introduction)

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