Histogram diffusion and its application to color histogram equalization

Abstract

In many image-processing applications the grayscale histogram equalization (GHE) is one of the simplest and most effective techniques for contrast enhancement. Various color histogram equalization (CHE) methods have been proposed to extend GHE for color images. Their performance shows large difference in terms of uniformity and computational time. The marginal (or conditional) CHE applies GHE directly to the marginal histogram of color image. Although these techniques provide fast and efficient algorithms they do not consider the correlation between different bands. In contrast, the multidimensional histogram equalization does not utilize the order information of histogram and can be extended into multi-dimension. In this thesis a new method called the “histogram diffusion” that extends the GHE method to arbitrary dimensions is proposed. Ranges in a histogram are specified as overlapping bars of uniform heights and variable widths which are proportional to their frequencies. This diagram is called the “vistogram.” As an alternative approach to GHE, the squared error of the vistogram from the uniform distribution is minimized. Bars in the vistogram are approximated by Gaussian functions. It allows to achieve desired smoothness and convexity of the squared error (the uniformity potential). The histogram diffusion principle is formulated as a nonlinear autonomous system of ordinary differential equations (ODEs). The histogram diffusion method possesses valuable properties including: i) dynamical and numerical stability, and ii) low computational complexity. The smooth and bounded uniformity potential has equilibrium points which guarantee dynamic stability. It bounds the mean squared error of a histogram and the squared error of a vistogram for uniformity at a fixed scale. Gaussian particles diffuse along the convex direction of the uniformity potential function. The continuity of mapping obtained by the histogram diffusion method is proven using Gronwall’s ineq...