Convergence of a finite volume scheme for a parabolic system applied to image processing

Abstract

We analyze a finite volume scheme for a nonlinear reaction-diffusion system applied to image processing. First, we demonstrate the existence of a solution to the finite volume scheme. Then, based on the derivation of a series of a priori estimates and the use of Kolmogorov’s compactness criterion, we prove that the solution to the finite volume scheme converges to the weak solution. In the numerical experiments, we show the effectiveness of the proposed model with respect to the modified (in the sense of Catté, Lions, Morel and Coll) Perona-Malik nonlinear image selective smoothing equation in terms of preserving small details, texture, and fine structures.

Keywords

• image processing
• perona-malik equation
• finite volume method
• convergence
• nonlinear reaction-diffusion
• 65m08
• 65m12
• 35k57
• 94a08