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

We present new results concerning the approximation of the total variation, $\int_Ω |\nabla u|$, of a function $u$ by non-local, non-convex functionals of the form $Λ_δu = \int_Ω \int_Ω \frac{δφ\big( |u(x) - u(y)|/ δ\big)}{|x - y|^{d+1}} \, dx \, dy,$ as $δ\to 0$, where $Ω$ is a domain in $\mathrm{R}^d$ and $φ: [0, + \infty) \to [0, + \infty)$ 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.

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