Basis beeldverwerking (8D040) dr. Andrea Fuster Prof.dr. Bart ter Haar Romeny Prof.dr.ir. Marcel Breeuwer dr. Anna Vilanova Histogram equalization

Contact dr. Andrea Fuster – 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(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

rkrk 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

Natural image histogram

Example histograms

More examples histograms

Mean Variance Histogram Features Mean: image mean intensity, measure of brightness Variance: measure of contrast

Questions? Any questions so far?

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 ② satisfied? Cumulative distribution function (CDF) Probability density function (PDF) p is always non- negative This means the cumulative probability function is monotonically increasing, ① ok!

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!