Wellposedness of a nonlocal nonlinear diffusion equation of image processing

TL;DR

This paper proves existence and uniqueness for a regularized version of the Perona-Malik image processing equation using advanced mathematical frameworks, ensuring well-posedness for non-smooth initial data.

Contribution

It introduces a novel functional setting based on singular Riemannian manifolds to analyze a degenerate nonlinear diffusion equation in image processing.

Findings

Established existence and uniqueness of solutions

Applied singular Riemannian manifold framework

Extended analysis to non-smooth initial data

Abstract

Existence and uniqueness are established for a degenerate regularization of the well-known Perona-Malik equation proposed by the first author for non-smooth initial data. The results heavily rely on the choice of appropriate functional setting inspired by a recent approach to degenerate parabolic equations via so-called singular Riemannian manifolds introduced by Herbert Amann.