Non-local Functionals Related to the Total Variation and Connections with Image Processing

We present new results concerning the approximation of the total variation, ∫ Ω| ∇ u| , of a function u by non-local, non-convex functionals of the form Λδ(u)=∫Ω∫Ωδφ(|u(x)-u(y)|/δ)|x-y|d+1dxdy,as δ→ 0 , where Ω is a domain in Rd and φ: [0 , + ∞) → [0 , + ∞) 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 Γ -convergence illuminates the situation, but also introduces mysterious novelties. The original motivation of our work comes from Image Processing. © 2018, Springer International Publishing AG, part of Springer Nature.