CV 431 Lec04 Histograms

Roger S. Gaborski 1

Introduction to Computer Vision

Lecture 4Dr. Roger S. Gaborski

Roger S. Gaborski 2

HW#2 Due 02/13

Roger S. Gaborski 3

In Class Exercise Review1. Given the following MATLAB code:>> image1 = rand([3])image1 = 0.9500 0.6555 0.0318 0.7431 0.1712 0.2769 0.3922 0.7060 0.0462 >> image2 = imadjust(image1, [ .1,.75],[.2, .6]) Carefully draw the transformation map specified by the imadjust statement. Label the x and y axis.

4Roger S. Gaborski

Chapter 3 www.prenhall.com/gonzalezwoodseddins

Brighter Output (gamma<1) Darker Output (gamma>1)

Gamma specifies the shape of the curve

Characteristics:• gamma >1 : all pixels become darker• gamma <1 : all pixels become brighter• gamma =1 : linear transform

5Roger S. Gaborski

image1 =

0.9500 0.6555 0.0318 0.7431 0.1712 0.2769 0.3922 0.7060 0.0462 >> image2 = imadjust(image1, [ .1,.75],[.2, .6])

image2 =

0.6000 0.5418 0.2000 0.5958 0.2438 0.3089 0.3798 0.5729 0.2000

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Example0

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image2 = imadjust(image1, [ .1,.75],[.2, .6],1)0

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image2 = imadjust(image1, [ .1,.75],[.2, .6],1)0

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Gray Scale Ramp Image

Roger S. Gaborski

Minimum gray value = .01Maximum gray value = 1.0

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Ramp Image

Roger S. Gaborski

image_ramp = zeros(100); for i = 1:100 image_ramp(:,i) = i*.01;end fprintf('minimum gray value = %d, \n', min(image_ramp(:)));fprintf('maximun gray value = %d, \n', max(image_ramp(:)));figure, plot(image_ramp(50,:)), xlabel('pixel position'), ylabel('pixel value') title('values')pause

figure, imshow(image_ramp),title('ramp intensity image')pause

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Map .01 to .5 and 1.0 to .75

Roger S. Gaborski

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disp('Use imadjust to map to .50 to .75');

image1 = imadjust(image_ramp,[.01, 1.0],[.5, 0.75]); %map to .5 to .75

fprintf('minimum gray value = %d, \n', min(image1(:)));fprintf('maximun gray value = %d, \n', max(image1(:)));

figure, plot(image1(50,:)), xlabel('pixel position'), ylabel('pixel value') title('values')axis([0,100,0,1])Grid

pause figure, imshow(image1, [0,1]);

pause

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Map .01 to .5 and 1.0 to .75

Roger S. Gaborski

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Map values in range <=.25 to .35 and >=.50 to .65');

Roger S. Gaborski

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Roger S. Gaborski

disp('Use imadjust to map values in range <=.25 to .35 and >=.50 to .65');image2 = imadjust(image_ramp,[.25, .50],[.35, 0.65]);fprintf('minimum gray value = %d, \n', min(image2(:)));fprintf('maximun gray value = %d, \n', max(image2(:)));figure, plot(image2(50,:)), xlabel('pixel position'), ylabel('pixel value') title('values')axis([0,100,0,1])grid pause figure, imshow(image2, [0,1]);

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Roger S. Gaborski

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Contrast Stretching Transformation• Creates an image with higher contrast than the input

image:

– r: intensities of input; – s: intensities of output;– m: threshold point (see graph) ; – E: controls slope.

Roger S. Gaborski

1( )1 ( )E

s T r mr

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E controls the slope of the function

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1( )1 ( )E

s T r mr

Input(r)

Output(s)

Introduction to Computer Vision

Lecture 4Dr. Roger S. Gaborski

22Roger S. Gaborski

117 83 59 59 68 77 84 94 82 67 62 70 83 86 85 81 71 65 77 89 86 82 76 67 72 90 97 86 66 54 68 104 121 107 85 46 58 89 138 165 137 91 38 80 147 200 211 187 138 40 80 149 197 202 187 146 56 76 114 159 181 160 113

An image shows the spatialdistribution of gray level values

We can summary this informationby only retaining the distributionif gray level values:

value count

1 0

40 8

…

…

104 32

181 45

255 0

PARTIAL IMAGE INFO:

Intensity image is simplya matrix of numbers

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Image Histogram

Roger S. Gaborski

Plot of Pixel Count as a Function of Gray Level Value

20 40 60 80 100 120 140 160 180 200 2200

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Gray Level Value

Pixel Count

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Histogram

• Histogram consists of– Peaks: high concentration of gray level values– Valleys: low concentration– Flat regions

Roger S. Gaborski

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Formally, Image Histograms

Histogram:• Digital image• L possible intensity levels in range [0,G]• Defined: h(rk) = nk

– Where rk is the kth intensity level in the interval [0,G] and nk is the number of pixels in the image whose level is rk .

– G: uint8 255 uint16 65535

double 1.0

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Notation

• L levels in range [0, G]• For example:• 0, 1, 2, 3, 4, in this case G = 4, L = 5

– Since we cannot have an index of zero,• In this example, index of:

Index 1 maps to gray level 02 maps to 13 maps to 24 maps to 35 maps to 4

Roger S. Gaborski

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Normalized Histogram

• Normalized histogram is obtained by dividing elements of h(rk) by the total number of pixels in the image (n):

for k = 1, 2,…, L

p(rk) is an estimate of the probability of occurrence of intensity level rk

Roger S. Gaborski

( )( ) ,k kk

h r np rn n

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MATLAB Histogram

• h = imhist( f, b )– h is the histogram, h(rk) – f is the input image– b is the number of bins (default is 256)

• Normalized histogram

Roger S. Gaborski

( , )( )

imhist f bpnumel f

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Color and Gray Scale Images

Roger S. Gaborski

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Background: Gray Image

Roger S. Gaborski

>> I = imread('Flags.jpg');>> figure, imshow(I) % uint8

>> Im = im2double(I); % convert to double>> Igray = (Im(:,:,1)+Im(:,:,2)+Im(:,:,3))/3;>> figure, imshow(Igray)

There is also the rgb2gray function that resultsin a slightly different image

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Gray Scale Histogram

Roger S. Gaborski

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Plots

• bar(horz, v, width)– v is row vector

• points to be plotted– horz is a vector same dimension as v

• increments of horizontal scale• omitted axis divided in units 0 to length(v)

– width number in [0 1]• 1 bars touch• 0 vertical lines• 0.8 default

Roger S. Gaborski

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p= imhist(Igray)/numel(Igray);>> h1 = p(1:10:256);>> horz = (1:10:256);>> figure, bar(horz,h1)

Review other examplesin text and in MATLABdocumentation

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Color and Gray Scale ImagesRecall from Previous Slide

Roger S. Gaborski

Gray Scale Histogram


Normalized Gray Scale Histogram


>> p= imhist(Igray)/numel(Igray);>> figure, plot(p)



imhist(Igray)/numel(Igray); imhist(Igray,32)/numel(Igray)

256 bins 32 bins



>> p= imhist(Igray)/numel(Igray);>> figure, plot(p)

Gray level values

prob

abili

ty


0 50 100 150 200 250 3000

0.005

0.01

0.015

0.02

0.025

0.03

0 50 100 150 200 250 3000

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0 50 100 150 200 250 3000

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Original Dark Light

Contract enhancement

• How could we transform the pixel values of an image so that they occupy the whole range of values between 0 and 255?


Gray Scale Transformation

• How could we transform the pixel values of an image so that they occupy the whole range of values between 0 and 255?

• If they were uniformly distributed between 0 and x we could multiply all the gray level values by 255/x

• BUT – what if they are not uniformly distributed??


Cumulative Distribution Function


Histogram CDF

Histogram Equalization(HE)

• HE generates an image with equally likely intensity values• Transformation function: Cumulative Distribution

Function (CDF)• The intensity values in the output image cover the full

range, [0 1]• The resulting image has higher dynamic range• The values in the normalized histogram are approximately

the probability of occurrence of those values


Histogram Equalization• Let pr(rj), j = 1, 2, … , L denote the histogram associated with

intensity levels of a given image

• Values in normalized histogram are approximately equal to the probability of occurrence of each intensity level in image

• Equalization transformation is:


k = 1,2,…,Lsk is intensity valueof output rk is input value

Sum of probability up to k value

1 1

( ) ( )k k

jk k r j

j j

ns T r p r

n

Histogram Equalization Example

• g = histeq(f, nlev) where f is the original image and nlev number of intensity levels in output image



Original ImageINPUT

Transformation


Input Gray Level Value

x255

Out

put G

ray

Leve

l Val

ue


OUTPUT

Equalization of Original Image


histogram Equalization of Light Image


histogram Equalization of Dark Image

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Simple Histogram Equalization Example

Roger S. Gaborski

Histogram Equalization


Input Image Output Image

Adaptive Equalization

• g = adapthisteq(f, parameters..)• Contrast-limited adaptive histogram equalization• Process small regions of the image (tiles) individually• Can limit contrast in uniform areas to avoid noise

amplification


55Roger S. Gaborski

>> help adapthisteq adapthisteq Contrast-limited Adaptive Histogram Equalization (CLAHE). adapthisteq enhances the contrast of images by transforming the values in the intensity image I. Unlike HISTEQ, it operates on small data regions (tiles), rather than the entire image. Each tile's contrast is enhanced, so that the histogram of the output region approximately matches the specified histogram. The neighboring tiles are then combined using bilinear interpolation in order to eliminate artificially induced boundaries. The contrast, especially in homogeneous areas, can be limited in order to avoid amplifying the noise which might be present in the image.

J = adapthisteq(I) Performs CLAHE on the intensity image I. J = adapthisteq(I,PARAM1,VAL1,PARAM2,VAL2...) sets various parameters. Parameter names can be abbreviated, and case does not matter. Each string parameter is followed by a value as indicated below: 'NumTiles' Two-element vector of positive integers: [M N]. [M N] specifies the number of tile rows and columns. Both M and N must be at least 2. The total number of image tiles is equal to M*N. Default: [8 8].

Adaptive Histogram Equalization


Grayscale Adaptive Histogram EqualizedGrayscale Histogram Equalized

Default, 8x8 tiles

Adaptive Equalization


Grayscale Adaptive Histogram Equalized

Number of Tiles: 24x24

Grayscale Adaptive Histogram Equalized

Number of Tiles: 24x24, ClipLimit = .05

Create a ‘color image’


>> red = rand(5)

red =

0.0294 0.0193 0.3662 0.7202 0.0302 0.7845 0.3955 0.2206 0.4711 0.2949 0.7529 0.1159 0.6078 0.9778 0.5959 0.1586 0.1674 0.5524 0.9295 0.1066 0.7643 0.6908 0.3261 0.5889 0.1359

>> green = rand(5)

green =

0.2269 0.5605 0.6191 0.0493 0.1666 0.0706 0.4051 0.3297 0.7513 0.6484 0.9421 0.0034 0.8243 0.7023 0.8097 0.8079 0.5757 0.6696 0.9658 0.8976 0.0143 0.3176 0.6564 0.1361 0.0754

>> blue = rand(5)

blue =

0.6518 0.0803 0.8697 0.6260 0.9642 0.5554 0.2037 0.8774 0.5705 0.6043 0.8113 0.8481 0.5199 0.0962 0.8689 0.5952 0.2817 0.6278 0.7716 0.8588 0.5810 0.9290 0.2000 0.1248 0.7606

First create three color planes of data


>> colorIm(:,:,1)=red;>> colorIm(:,:,2)=green;>> colorIm(:,:,3)=blue;>> colorIm

figure imshow(colorIm, 'InitialMagnification', 'fit')

colorIm(:,:,1) =

0.0294 0.0193 0.3662 0.7202 0.0302 0.7845 0.3955 0.2206 0.4711 0.2949 0.7529 0.1159 0.6078 0.9778 0.5959 0.1586 0.1674 0.5524 0.9295 0.1066 0.7643 0.6908 0.3261 0.5889 0.1359

colorIm(:,:,2) =

0.2269 0.5605 0.6191 0.0493 0.1666 0.0706 0.4051 0.3297 0.7513 0.6484 0.9421 0.0034 0.8243 0.7023 0.8097 0.8079 0.5757 0.6696 0.9658 0.8976 0.0143 0.3176 0.6564 0.1361 0.0754

colorIm(:,:,3) =

0.6518 0.0803 0.8697 0.6260 0.9642 0.5554 0.2037 0.8774 0.5705 0.6043 0.8113 0.8481 0.5199 0.0962 0.8689 0.5952 0.2817 0.6278 0.7716 0.8588 0.5810 0.9290 0.2000 0.1248 0.7606

colorIm


colorIm(1,1,: )

colorIm(4,4,: )


colorIm(:,:,1) =

0.0294 0.0193 0.3662 0.7202 0.0302 0.7845 0.3955 0.2206 0.4711 0.2949 0.7529 0.1159 0.6078 0.9778 0.5959 0.1586 0.1674 0.5524 0.9295 0.1066 0.7643 0.6908 0.3261 0.5889 0.1359

colorIm(:,:,2) =

0.2269 0.5605 0.6191 0.0493 0.1666 0.0706 0.4051 0.3297 0.7513 0.6484 0.9421 0.0034 0.8243 0.7023 0.8097 0.8079 0.5757 0.6696 0.9658 0.8976 0.0143 0.3176 0.6564 0.1361 0.0754

colorIm(:,:,3) =

0.6518 0.0803 0.8697 0.6260 0.9642 0.5554 0.2037 0.8774 0.5705 0.6043 0.8113 0.8481 0.5199 0.0962 0.8689 0.5952 0.2817 0.6278 0.7716 0.8588 0.5810 0.9290 0.2000 0.1248 0.7606

• What are two methods to convert from a color image to a gray scale image?


RECALL

• What are two methods to convert from a color image to a gray scale image?– Average red, green and blue pixels


Average

• For example:


>> colorImAverage = ( colorIm(:,:,1) + colorIm(:,:,2) + colorIm(:,:,3) )/3

colorImAverage =

0.3027 0.2200 0.6183 0.4651 0.3870 0.4701 0.3348 0.4759 0.5976 0.5159 0.8354 0.3224 0.6507 0.5921 0.7582 0.5206 0.3416 0.6166 0.8890 0.6210 0.4532 0.6458 0.3942 0.2833 0.3240

>> figure, imshow(colorImAverage, 'InitialMagnification', 'fit')

Gray scale version of color image


.5976

.5921

Color and Gray scale Images


Color and Gray scale Images


Conversion to gray scale results in a loss of information

• What are two methods to convert from a color image to a gray scale image?– Average red, green and blue pixels– Matlab’s rgb2gray function


MATLAB’s rgb2gray Function


>> colorIm_rgb2gray = rgb2gray(colorIm)

colorIm_rgb2gray =

0.2163 0.3439 0.5721 0.3156 0.2168 0.3393 0.3792 0.3596 0.6469 0.5377 0.8706 0.1333 0.7249 0.7155 0.7525 0.5895 0.4202 0.6298 0.9328 0.6567 0.3031 0.4989 0.5056 0.2702 0.1716

colorIm and rgb2gray(colorIm)


How does rgb2gray work?


rgb2gray converts RGB values to grayscale values by forming a weighted sum of the R, G, and B components:

Gray = 0.2989 * R + 0.5870 * G + 0.1140 * B

Color and Gray Scale Images


CV 431 Lec04 Histograms

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