Histogram Modification via Differential Equations

Abstract

The explicit use of partial differential equations (PDEs) in image processing became a major research topic in the past years. In this work we present a framework for histogram (pixel-value distribution) modification via ordinary and partial differential equations. In this way, the image contrast is improved. We show that the histogram can be modified to achieve any given distribution as the steady state solution of an image flow. The contrast modification can be performed while simultaneously reducing noise in a unique PDE, avoiding noise sharpening effects of classical algorithms. The approach is extended to local contrast enhancement as well. A variational interpretation of the flow is presented and theoretical results on the existence of solutions are given.1997 Academic Press

An Algorithmic Analysis of Variational Models for Perceptual Local Contrast Enhancement

“…One of the main inspirations of paper [7] was the following result about the variational interpretation of histogram equalization [11].…”

“…The main result of paper [11] is that, if I * = argmin I E hist eq (I), then I * has equalized histogram, i.e. all the intensity levels have the same occurrence probability in the image.…”

“…all the intensity levels have the same occurrence probability in the image. In [11], the authors prove this result by showing that, if we denote by H the cumulative histogram, then the image function I * satisfies the equation H(I * (x)) = I * (x), for all x ∈ Ω, which implies the flatness of the histogram of I * , see e.g. [4].…”

“…The minimization of E hist eq (I) can be achieved through gradient descent. With respect to this, another important result of [11] is that, if I 0 represents any input image, then the initial value problem…”

Duality Principle for Image Regularization and Perceptual Color Correction Models

“…Regarding contrast modification, the pioneer variational approach is due to Sapiro and Caselles [23], who performed contrast enhancement for histogram equalization purpose. Since then, this variational formulation has been generalized and applied in different contexts: perceptual color correction [4], [20], tone mapping [14], and gamut mapping [25] to name a few.…”