On ill-posed anisotropic diffusion models

This paper describes a class of ill-posed anisotropic diffusion models of the type presented by Perona and Malik (1990). The analysis is based on a previous result that anisotropic diffusion is a steepest descent motion on an energy surface and its behavior is thus determined by the shape of this energy surface. We show that the class of diffusion models are ill-posed because the energy surface is discontinuous at all continuous images, and all step images, which are dense in the space of piecewise images, are global minima of the energy surface