Scale-space Properties of Nonlinear Diiusion Filtering with a Diiusion Tensor

In spite of its lack of theoretical justiication, nonlinear diiusion ltering 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 diiusion ltering as a scale-space transformation which is exible enough to simplify images without loosing the capability of enhancing edges. By studying the Lyapunov functionals, it is shown that nonlinear diiusion reduces L p norms and central moments and increases the entropy of images. The proposed anisotropic class utilizes a diiusion 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 satisses 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. .