BEHAVIORAL-ANALYSIS OF ANISOTROPIC DIFFUSION IN IMAGE-PROCESSING

In this paper, we analyze the behavior of the anisotropic diffusion mo del of Perona and Malik. The main idea is to express the anisotropic d iffusion equation as coming from a certain optimization problem, so it s behavior can be analyzed based on the shape of the corresponding ene rgy surface. We show that anisotropic diffusion is the steepest descen t method for solving an energy minimization problem. It is demonstrate d that an anisotropic diffusion is well posed when there exists a uniq ue global minimum for the energy functional and that the ill posedness of a certain anisotropic diffusion is caused by the fact that its ene rgy functional has an infinite number of global minima that are dense in the image space. We give a sufficient condition for an anisotropic diffusion to be well posed and a sufficient and necessary condition fo r it to be ill posed due to the dense global minima. The mechanism of smoothing and edge enhancement of anisotropic diffusion is illustrated through a particular orthogonal decomposition of the diffusion operat or into two parts: one that diffuses tangentially to the edges and the refore acts as an anisotropic smoothing operator, and the other that f lows normally to the edges and thus acts as an enhancement operator.