In this paper, we analyze the behavior of the anisotropic diffusion model of Perona and Malik. The main idea is to express the anisotropic diffusion equation as coming from a certain optimization problem, so its behavior can be analyzed based on the shape of the corresponding energy surface. We show that anisotropic diffusion is the steepest descent method for solving an energy minimization problem. It is demonstrated that an anisotropic diffusion is well posed when there exists a unique global minimum for the energy functional and that the ill posedness of a certain anisotropic diffusion is caused by the fact that its energy 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 for 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 operator into two parts: one that diffuses tangentially to the edges and therefore acts as an anisotropic smoothing operator, and the other that flows normally to the edges and thus acts as an enhancement operator.
Bibliographical noteFunding Information:
Manuscript received February 6, 1995; revised March 8, 1996. This work was supported by the BMDOAST program managed by the Office of Naval Research under Contract N00014-92-J-1911, by the National Science Foundation under Contract ECS-9122106, by the Air Force Office of Scientific Research under Contract F49620-94-1-0058DEF, and by the Army Research Office under Contracts DAAH04-94-G-0054 and DAAH04-93-C-0332. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Dan Schonfeld. The authors are with the Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: firstname.lastname@example.org). Publisher Item Identifier S 10.57-7149(96)07893-I.