Non-local Functionals Related to the Total Variation and Connections with 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 $$\begin{aligned} \Lambda _\delta (u) = \int _{\Omega } \int _{\Omega } \frac{\delta \varphi \big ( |u(x) - u(y)|/ \delta \big )}{|x - y|^{d+1}} \, dx \, dy, \end{aligned}$$ as $$\delta \rightarrow 0$$ , where $$\Omega $$ is a domain in $$\mathbb {R}^d$$ and $$\varphi : [0, + \infty ) \rightarrow [0, + \infty )$$ 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.