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The Essential Guide to Image Processing- P3:We are in the middle of an exciting period of time in the field of image processing. Indeed, scarcely a week passes where we do not hear an announcement of some new technological breakthrough in the areas of digital computation and telecommunication. | 58 CHAPTER 3 Basic Gray Level Image Processing for all n. Since Pf is a continuous function 3.26 - 3.28 represents a smooth mapping of the histogram of image f to an image with a smooth histogram. At first 3.27 may seem confusing since the function Pf that is computed from f is then applied to f .To see that a flat histogram is obtained we use the probabilistic interpretation of the histogram. The cumulative histogram of the resulting image g is Pg x Pr g x Pr Pf f x Pr f P x Pf Pf1 x x 3.29 for 0 x 1. Finally the normalized histogram of g is pg x dPg x dx 1 3.30 for 0 x 1. Since pg x is defined only for 0 x 1 FSHS in 3.26 is required to stretch the flattened histogram to fill the grayscale range. To flatten the histogram of a digital image f first compute the discrete cumulative normalized histogram Pf k apply 3.28 at each n then 3.26 to the result. However while an image with a perfectly flat histogram is the result in the ideal continuous case outlined above in the digital case the output histogram is only approximately flat or more accurately flatter than the input histogram. This follows since 3.26 - 3.28 collectively is a point operation on the image f so every occurrence of gray level k maps to Pf k in g. Hence histogram bins are never reduced in amplitude by 3.26 - 3.28 although they may increase if multiple gray levels map to the same value thus destroying information . Hence the histogram cannot be truly equalized by this procedure. Figures 3.14 and 3.15 show histogram equalization applied to our ongoing example images students and books respectively. Both images are much more striking and viewable than the original. As can be seen the resulting histograms are not really flat it is flatter in the sense that the histograms are spread as much as possible. However the heights of peaks are not reduced. As is often the case with expansive point operations FIGURE 3.14 Histogram equalization applied to the image students. 3.5 Nonlinear Point Operations on Images
