Images are captured at low contrast in a number of different scenarios. The main reason for this problem is poor lighting conditions (e.g., pictures taken at night or against the Sun's rays). As a result, the image is too dark or too bright and is inappropriate for visual inspection or simple observation. The most common way to improve the contrast of an image is to modify its pixel value distribution, or histogram. A schematic example of the contrast enhancement problem and its solution by means of histogram modification is given in Fig. 7.1. On the left, we see a low-contrast image with two different squares, one inside the other, and its corresponding histogram.We can observe that the image has low contrast and the different objects cannot be identified, as the two regions have almost identical gray values. On the right we see what happens when we modify the histogram in such a way that the gray values corresponding to the two regions are separated. The contrast is improved immediately. An additional example, this time for a real image, is given in Fig. 7.2.

In this chapter, we first follow Ref. [339, 340] and show show how to obtain any gray-level distribution as the steady state of an ODE and present examples for different pixel value distributions. Uniform distributions are usually used in most contrast enhancement applications [160]. On the other hand, for specific tasks, the exact desirable distribution can be dictated by the application, and the technique presented here applies as well.