Scale-Space Properties of Nonlinear Diffusion Filtering with a Diffusion Tensor

In spite of its lack of theoretical justification, nonlinear diffusion filtering has become a powerful image enhancement tool in recent years. The goal of the present paper is to provide a mathematical foundation for continuous nonlinear diffusion filtering as a scale-space transformation which is flexible enough to simplify images without loosing the capability of enhancing edges. By studying the Lyapunov functionals, it is shown that nonlinear diffusion reduces L p norms and central moments and increases the entropy of images. The proposed anisotropic class utilizes a diffusion tensor which may be adapted to the image structure. It permits existence, uniqueness and regularity results, the solution depends continuously on the initial image, and it satisfies an extremum principle. All considerations include linear and certain nonlinear isotropic models and apply to m- dimensional vector-valued images. The results are juxtaposed to linear and morphological scale-spaces. . Keywords....