Cross-Diffusion Systems for Image Processing: II. The Nonlinear Case

Abstract

In this paper we study the application of 2×2 nonlinear cross-diffusion systems as mathematical models of image filtering. These are systems of two nonlinear, coupled partial differential equations of parabolic type. The nonlinearity and cross-diffusion character are provided by a nondiagonal matrix of diffusion coefficients that depends on the variables of the system. We prove the well-posedness of an initial-boundary-value problem with Neumann boundary conditions and uniformly positive definite cross-diffusion matrix. Under additional hypotheses on the coefficients, the models are shown to satisfy the scale-space properties of shift, contrast, average grey and translational invariances. The existence of Lyapunov functions and the asymptotic behaviour of the solutions are also studied. According to the choice of the cross-diffusion matrix (on the basis of the results on filtering with linear cross-diffusion, discussed by the authors in a companion paper and the use of edge stopping functions ) the performance of the models is compared by computational means in a filtering problem. The numerical results reveal differences in the evolution of the filtering as well as in the quality of edge detection given by one of the components of the system, in terms of the cross-diffusion matrix.