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Histogram Equalization - Home | UCI Mathematics

Histogram EqualizationHistogram Equalization is a technique for adjusting image intensities to enhance a given image represented as amrbymcmatrix of integer pixel intensities rangingfrom 0 toL the number of possible intensity values, often 256. Letpdenote thenormalized Histogram offwith a bin for each possible intensity. Sopn=number of pixels with intensityntotal number of pixelsn= 0,1, .., L Histogram equalized imagegwill be defined bygi,j= floor((L 1)fi,j n=0pn),(1)where floor() rounds down to the nearest integer. This is equivalent to transforming thepixel intensities,k, offby the functionT(k) = floor((L 1)k n=0pn).The motivation for this transformation comes from thinkingof the intensities offandgascontinuous random variablesX,Yon [0, L 1] withYdefined byY=T(X) = (L 1) X0pX(x)dx,(2)wherepXis the probability density function the cumulative distributive functionofXmultiplied by (L 1). Assume for simplicity thatTis differentiable and invertible.

number of pixels with intensity n total number of pixels n = 0,1,...,L− 1. The histogram equalized image g will be defined by g i,j = floor((L− 1) Xfi,j n=0 p n), (1) where floor() rounds down to the nearest integer. This is equivalent to transforming the pixel intensities, k, of f by the function T(k) = floor((L− 1) Xk n=0 p n).

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