Non-local functionals related to the total variation and connections with Image Processing

TL;DR

This paper investigates how certain non-local, non-convex functionals approximate the total variation of functions, with implications for image processing, highlighting delicate convergence modes and open problems.

Contribution

It introduces new results on the approximation of total variation by non-local functionals and explores the complex convergence behavior using Gamma-convergence, with applications to image processing.

Findings

Established conditions for the approximation of total variation by non-local functionals.

Analyzed the delicate mode of convergence and identified open problems.

Connected the theoretical framework to practical applications in image processing.

Abstract

We present new results concerning the approximation of the total variation, $\int_{\Omega} |\nabla u|$, of a function $u$ by non-local, non-convex functionals of the form $$ \Lambda_\delta u = \int_{\Omega} \int_{\Omega} \frac{\delta \varphi \big( |u(x) - u(y)|/ \delta\big)}{|x - y|^{d+1}} \, dx \, dy, $$ as $\delta \to 0$, where $\Omega$ is a domain in $\mathrm{R}^d$ and $\varphi: [0, + \infty) \to [0, + \infty)$ is a non-decreasing function satisfying some appropriate conditions. The mode of convergence is extremely delicate and numerous problems remain open. De Giorgi's concept of Gamma-convergence illuminates the situation, but also introduces mysterious novelties. The original motivation of our work comes from Image Processing.