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 – [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 Histogram MN: total number of pixels (image of dimensions MxN) rk What do the histograms of these images look like? Bimodal histogram Tri- (or more) modal histogram Example histograms More examples histograms More examples histograms Histogram Features • Mean • Variance Mean: image mean intensity, measure of brightness Variance: 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 ② satisfied? • Cumulative distribution function (CDF) • Probability density function (PDF) p is always nonnegative • 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!