In spite of its lack of theoretical justification, nonlinear diffusion filtering has become a powerful image enhancement tool in the recent years. The goal of the present paper is to provide a mathematical foundation for nonlinear diffusion filtering as a scale-space transformation which is flexible enough to simplify images without loosing the capability of enhancing edges. By stuying the Lyapunow functional, it is shown that nonlinear diffusion reduces Lp 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 fulfills 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.
The edge enhancement property of a nonlinear diffusion equation with a suitable expression for the diffusivity is an important feature for image processing. We present an algorithm to solve this equation in a wavelet basis and discuss its one dimensional version in some detail. Sample calculations demonstrate principle effects and treat in particular the case of highly noise perturbed signals. The results are discussed with respect to performance, efficiency, choice of parameters and are illustrated by a large number of figures. Finally, a comparison with a Fourier method and a finite volume method is performed.