Nonlinear diffusion: A probabilistic view

Abstract

A probabilistic view of diffusion is presented. A discrete symmetric random walk is shown to be equivalent to a heat equation evolution, and an extension to nonlinear evolutions including Perona-Malik equation is shown to be of utmost importance for analysis. Upon unraveling the limitations as well as the advantages of such an equation, we are able to propose a new approach which is demonstrated to outperform existing approaches, and to lift the long-standing problem of when to stop the evolution. Substantiating examples of image enhancement and segmentation are provided.