# Basis

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Basis beeldverwerking (8D040)dr. Andrea FusterProf.dr. Bart ter Haar RomenyProf.dr.ir. Marcel Breeuwerdr. Anna Vilanova

Histogram equalization

Slide2Contact

d

r. Andrea Fuster –

A.Fuster@tue.nl

Mathematical image analysis at W&I and Biomedical image analysis at BMT

HG 8.84 / GEM-Z 3.108

Slide3Today

Definition of histogram

Examples

Histogram features

Histogram equalization:

Continuous case

Discrete case

Examples

Slide4Histogram definition

Histogram is a discrete function

h

(

r

k

)

=

N(

r

k

)

,

where

r

k

is the

k-th

intensity value, and

N(

r

k

)

is the number of pixels with intensity

r

k

Histogram normalization by dividing

N(

r

k

)

by the number of pixels in the image (MN)

Normalization turns histogram into a

probability distribution function

Slide5r

k

Histogram

MN: total number of pixels (image of dimensions

MxN

)

Slide6What do the histograms of these images look like?

Slide7Bimodal histogram

Slide8Tri- (or more) modal histogram

Slide9Natural image histogram

Slide10Example histograms

Slide11More examples histograms

Slide12More examples histograms

Slide13MeanVariance

Histogram Features

Mean: image mean intensity, measure of brightness

Variance: measure of contrast

Slide14Questions?

Any questions so far?

Slide15Histogram processing

Slide16Histogram processing

Slide17Histogram equalization

Idea: spread

the intensity values to cover the whole gray scale

Result: improved/increased contrast!☺

Slide18Histogram equalization – cont. case

Assume r is the intensity in an image with L levels:Histogram equalisation is a mapping of the formwith r the input gray value and s the resulting or mapped value

Slide19Histogram equalization – cont. case

Assumptions / conditions:① is monotonically increasing function in ②Make sure output range equal to input range

Slide20Histogram equalization – cont. case

Monotonically increasing function T(r)

Slide21Histogram equalization – cont. case

Consider a candidate function for T(r) – conditions ① and ② satisfied? Cumulative distribution function (CDF)Probability density function (PDF) p is always non-negativeThis means the cumulative probability function is monotonically increasing, ① ok!

Slide22Histogram equalization – cont. case

Does the CDF fit the second assumption? To have the same intensity range as the input image, scale with (L-1)

So ② ok!

Slide23Histogram equalization – cont. case

What happens when we apply the transformation function T(r) to the intensity values? – how does the histogram change?

Slide24Histogram equalization – cont. case

What is the resulting probability distribution?From probability theory

Slide25Histogram equalization – cont. case

Uniform:

What does this mean?

Slide26Histogram equalization – disc. case

Spreads the intensity values to cover the whole gray scale (improved/increased contrast)Fully automatic method, very easy to implement:

Slide27Histogram

equalization – disc. case

Notice something??

Slide28Demo of equalization in Mathematica

Original image

Original histogram

Transformation function T(r)

“

Equalised” image

“

Equalised

” histogram

Slide29End of part 1

And now we deserve a break!

Slide30Slide31

Slide32

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