Histogram equalization

Basis beeldverwerking (8D040)

dr. Andrea FusterProf.dr. Bart ter Haar RomenyProf.dr.ir. Marcel Breeuwerdr. Anna Vilanova

Histogram equalization

Contact

• dr. Andrea Fuster – [email protected]• Mathematical image analysis at W&I and Biomedical

image analysis at BMT • HG 8.84 / GEM-Z 3.108

Today

• Definition of histogram • Examples • Histogram features• Histogram equalization:

• Continuous case• Discrete case

• Examples

Histogram definition

• Histogram is a discrete function h(rk) = N(rk) , where

• rk is the k-th intensity value, and• N(rk) is the number of pixels with intensity rk

• Histogram normalization by dividing N(rk) by the number of pixels in the image (MN)

• Normalization turns histogram into a probability distribution function

rk

Histogram

MN: total number of pixels (image of dimensions MxN)

What do the histograms of these images look like?

Bimodal histogram

Tri- (or more) modal histogram

Example histograms

More examples histograms

More examples histograms

• Mean

• Variance

Histogram Features

Mean: image mean intensity, measure of brightnessVariance: measure of contrast

Questions?

• Any questions so far?

Histogram processing

Histogram processing

Histogram equalization

• Idea: spread the intensity values to cover the whole gray scale

• Result: improved/increased contrast!☺

Histogram equalization – cont. case

• Assume r is the intensity in an image with L levels:

• Histogram equalisation is a mapping of the form

• with r the input gray value and s the resulting or mapped value

Histogram equalization – cont. case

• Assumptions / conditions:• ① is monotonically increasing function in • ②

• Make sure output range equal to input range

Histogram equalization – cont. case

• Monotonically increasing function T(r)

Histogram equalization – cont. case

• Consider a candidate function for T(r) – conditions ① and satisfied?②

• Cumulative distribution function (CDF)• Probability density function (PDF) p is always non-

negative• This means the cumulative probability function is

monotonically increasing, ok!①

Histogram 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!

Histogram equalization – cont. case

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

Histogram equalization – cont. case

• What is the resulting probability distribution?• From probability theory

Histogram equalization – cont. case

• Uniform:

• What does this mean?

Histogram equalization – disc. case

• Spreads the intensity values to cover the whole gray scale (improved/increased contrast)

• Fully automatic method, very easy to implement:

Histogram equalization – disc. case

Notice something??

Demo of equalization in Mathematica

Original image

Original histogram

Transformation function T(r)

“Equalised” image

“Equalised” histogram

End of part 1

• And now we deserve a break!