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DREAMDREAM

PLANPLANIDEAIDEA

IMPLEMENTATIONIMPLEMENTATION

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Introduction to Image ProcessingIntroduction to Image Processing

Dr. Kourosh KianiEmail: kkiani2004@yahoo.comEmail: Kourosh.kiani@aut.ac.irEmail: Kourosh.kiani@semnan.ac.irWeb: www.kouroshkiani.com

Present to:Amirkabir University of Technology (Tehran

Polytechnic) & Semnan University

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Lecture 02

Histogram

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THE IMAGE HISTOGRAM

H(k) specifies the # of pixels with gray-value k

Let N be the number of pixels: P(k) = H(k)/N defines the normalized

histogram defines the accumulated

histogram

Occurrence(# of pixels)

Gray Level

k

kHN )(

k

i

iHkC1

)()(

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0.2

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1

gray level

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gray level

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gray level

Histogram

Normalized Histogram

Accumulated Histogram

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The image histogram does not fully represent the image

P(I)

I

P(I)

I

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0.5

EXAMPLES

H(I)

I

0.1

H(I)

I

0.1

Pixel permutation of the left image

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EXAMPLE

2 3 3 2

4 2 4 3

3 2 3 5

2 4 2 4

4x4 image

Gray scale = [0,9]histogram

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No. of pixels

Gray level

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Gray Level(j)

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No. of pixels 0 0 6 5 4 1 0 0 0 0

0 0 6 11 15 16 16 16 16 16

0 06

/16

11 /16

15 /16

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16 /16

s x 9 0 03.33

6.16

8.48

9 9 9 9 9

k

jjn

0

k

j

j

n

ns

0

EXAMPLE

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Grayscale Histogram

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WHAT IS HISTOGRAM ?

(3 ,8 ,5)

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HISTOGRAM

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2 1 0 0 2

5 2 0 0 4

1 1 2 4 1 0

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image histogram

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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

value count

1 0

40 8

…

…

104 32

181 45

255 0

PARTIAL IMAGE INFO:

HISTOGRAM

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IMAGE HISTOGRAM

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Plot of Pixel Count as a Function of Gray Level Value

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IMAGE HISTOGRAMS

The histogram of an image shows us the distribution of grey levels in the imageMassively useful in image processing, especially in segmentation

Grey Levels

Fre

quen

cies

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0.005

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0.015

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Original Dark Light

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HISTOGRAM

An image has low contrast when the complete range of possible values is not used.  Inspection of the histogram shows this lack of contrast.

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HISTOGRAM

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rgbI = imread('kourosh.jpg');grayI = rgb2gray(rgbI);%get image histogram[counts bins]=imhist(grayI); %show histogrambar(bins,counts);

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rgbI = imread('kouroshdark.jpg');grayI = rgb2gray(rgbI);%get image histogram[counts bins]=imhist(grayI); %show histogrambar(bins,counts);

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rgbI = imread('kouroshbright.jpg');grayI = rgb2gray(rgbI);%get image histogram[counts bins]=imhist(grayI); %show histogrambar(bins,counts);

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I = imread('pic01.png');%get image histogram[counts bins]=imhist(I); %show histogrambar(bins,counts);

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I = imread('pic03.png');%get image histogram[counts bins]=imhist(I); %show histogrambar(bins,counts);

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HISTOGRAM EQUALIZATION

The image shows the spatial distribution of gray values.

The image histogram discards the spatial information and shows the relative frequency of occurrence of the gray values.

0 3 3 2 5 5

1 1 0 3 4 5

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0 2 .05 1 2 .05 2 4 .11 3 6 .17 4 7 .20 5 8 .22 6 6 .17 7 1 .03

Image CountGray Value

Rel. Freq.

Sum= 36 1.00

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The histogram typically plots the absolute pixel count as a function of gray value:

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Pix

el C

ount

Gray Value

For an image with dimensions M by N

MNiHI

Ii

min

min

)(I max

HISTOGRAM EQUALIZATION

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PROBABILITY INTERPRETATION

The graph of relative frequency of occurrence as a function of gray value is also called a histogram:

Interpreting the relative frequency histogram as a probability distribution,

then:

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0.05

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Rela

tive F

requency

Gray Value

P(I(x,y) = i) = H(i)/(MxN)

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CUMULATIVE DENSITY FUNCTION Interpreting the relative frequency

histogram as a probability distribution, then:

Curve is called the cumulative distribution function

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Rela

tive F

requency

Gray Value

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1

Gray Value

P(I(x,y) = i) = H(i)/(MxN)

i

k

kPiQ0

)()(

i

k

kHiCH0

)()(

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Image histograms consist of peaks, valleys, and low plains

Peaks = many pixels concentrated in a few grey levels

Plains = small number of pixels distributed over a wider range of grey levels

HISTOGRAM EQUALIZATION

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HISTOGRAM EQUALIZATION The goal is to modify the gray levels of an image so

that the histogram of the modified image is flat. Expand pixels in peaks over a wider range of

gray-levels. “Squeeze” low plains pixels into a narrower

range of gray levels. Utilizes all gray values equally Example Histogram:

Note low utilization of small gray values

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200400600800

100012001400160018002000

Pix

el C

ount

Gray Value

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DESIRED HISTOGRAM

All gray levels are used equally. Has a tendency to enhance contrast.

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Pix

el C

ount

Gray Value

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BRUTE FORCE

0 0 256 22 from 7, 234 from 8 1 0 256 1 from 8, 255 from 9 2 0 256 195 from 9, 61 from 10 3 0 256 256 from 10 4 0 256 256 from 10 5 0 256 256 from 10 6 0 256 256 from 10 7 22 256 256 from 10 8 235 256 256 from 10 9 450 256 256 from 10

10 1920 256 67 from 10, 189 from 11 11 212 256 23 from 11, 10 from 12, 223 from 13 12 10 256 10 from 13, 246 from 14 13 233 256 256 from 14 14 672 256 170 from 14, 86 from 15 15 342 256 256 from 15

Gray Actual Desired How to get it Scale Count Count

Sum 4096 4096

How are the gray levels in the original image changed to produce the enhanced image?

Method 1. Choose points randomly.

Method 2. Choice depends on the graylevels of their neighboring points.

Computationally expensive.

Approximations.

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HISTOGRAM EQUALIZATION Mapping from one set of grey-levels, I, to a new

set, O. Ideally, the number of pixels, Np, occupying each

grey level should be:

To approximate this, apply the transform

Where CH is the cumulative histogram (see next slide)

j is the gray value in the source image i is the gray value in the equalized image

Np = M*N

G

i = MAX 0 ,roundCH(j)

Np

-1

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EXAMPLE

800 600 400 200 0

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ideal

800 600 400 200 0

G=8MxN=2400Np=300

j H(j) CH(j) i

0 100 01 800 22 700 43 500 64 100 65 100 76 100 77 0 7

100900160021002200230024002400

CH(j) = S H(i)i=0

j

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EXAMPLErgbI = imread('ship1.png');J = histeq(rgbI);imshow(rgbI)figure, imshow(J)%get image histogram[counts bins]=imhist(rgbI);%show histogramfigure(3)bar(bins,counts);%get image histogram[counts bins]=imhist(J); %show histogramfigure(4)bar(bins,counts);

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EXAMPLE

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Histogram Equalization: Example

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EXAMPLE

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COLOR IMAGE

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clc; % Clear the command window.close all; % Close all figures (except those of imtool.)imtool close all; % Close all imtool figures.clear; % Erase all existing variables.workspace; % Make sure the workspace panel is showing.fontSize = 20; % Read in standard MATLAB color demo image.rgbImage = imread('peppers.png');[rows columns numberOfColorBands] = size(rgbImage);figure (1)imshow(rgbImage, []); redPlane = rgbImage(:, :, 1);greenPlane = rgbImage(:, :, 2);bluePlane = rgbImage(:, :, 3); % Let's get its histograms.[pixelCountR grayLevelsR] = imhist(redPlane);figure(2)%show histogrambar(grayLevelsR,pixelCountR,'r');xlim([0 grayLevelsR(end)]); % Scale x axis manually.title('Histogram of red plane', 'Fontsize', fontSize);grid on;[pixelCountG grayLevelsG] = imhist(greenPlane);figure(3)%show histogrambar(grayLevelsG,pixelCountG,'g');xlim([0 grayLevelsG(end)]); % Scale x axis manually.title('Histogram of green plane', 'Fontsize', fontSize);grid on;[pixelCountB grayLevelsB] = imhist(bluePlane);figure(4)%show histogrambar(grayLevelsB,pixelCountB,'b');xlim([0 grayLevelsB(end)]); % Scale x axis manually.title('Histogram of blue plane', 'Fontsize', fontSize);grid on;

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Histogram of green plane

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Histogram of blue plane

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Histogram of red plane

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Questions? Discussion? Suggestions?

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