histogram theory

8/4/2019 Histogram Theory

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By

Debashree GhoshAsst. Prof./BME

ALPHA COLLEGE OF ENGINEERINGThirumazhisai, Chennai- 124


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Introduction

Increases the global contrast of images

Especially when the usable data of the image

is represented by close contrast values.

This allows for areas of lower local contrast to

gain a higher contrast.


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Histogram

Consider a image {x} and let nk be the numberof occurrences of gray level rk.

The probability of an occurrence of a pixel oflevel kin the image is

Also called normalized histogram Note: Sum of all components of the normalized histogram is

equal to 1


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Histogram Equalization

For any r transformed pixel value

s=T(r) 0


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Contd

ps(s) and pr(r) are PDF of random variables s

and r, respectively.

s is determined by gray level PDF of original

image dummy variable of integration

Right side of the equation if cumulative

distribution function Since PDFs are +ve , it follows that this transformation is

singled valued and monotonically increasing

Similarly Integral of PDF is in the range of [0,1]


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Contd.

We can find the value of ps (s)

0


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Contd

Discrete Formulation


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Original Image


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Value Count Value Count Value Count Value Count Value Count

52 1 64 2 72 1 85 2 113 1

55 3 65 3 73 2 87 1 122 1

58 2 66 2 75 1 88 1 126 1

59 3 67 1 76 1 90 1 144 1

60 1 68 5 77 1 94 1 154 1

61 4 69 3 78 1 104 2

62 1 70 4 79 2 106 1

63 2 71 2 83 1 109 1


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Value cdf Value cdf Value cdf Value cdf Value cdf

52 1 64 19 72 40 85 51 113 60

55 4 65 22 73 42 87 52 122 61

58 6 66 24 75 43 88 53 126 62

59 9 67 25 76 44 90 54 144 63

60 10 68 30 77 45 94 55 154 64

61 14 69 33 78 46 104 57

62 15 70 37 79 48 106 58

63 17 71 39 83 49 109 59


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The general histogram equalization formula is:

Where cdfmin is the minimum value of the

cumulative distribution function (in this case 1)

For example, the cdf of 78 is 46:


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Equalized values


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Example


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Contd..


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Histogram Matching

The Enhancement on a uniform histogram is

not the best approach

Use specific shape of histogram

Histogram matching also called as hostogram

spacification.


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Contd

r and z are continuous gray levels

pr(r) and pz (z) denotes correspondingcontinuous probability density functions.

Step1:

Step2:

Step3:


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Discrete Gray Level

Gjhk


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Original Image and its Histogram


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Histogram equalization


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histogram theory

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