Non-convex, non-local functionals converging to the total variation

We present new results concerning the approximation of the total variation, integral(Omega)vertical bar del u vertical bar, of a function u by non-local, non-convex functionals of the form Lambda delta(u) = integral(Omega)integral(Omega)delta phi(vertical bar u(x) - u(y)vertical bar/delta)/vertical bar x-y vertical bar(d+1)dxdy, as delta -> 0, where Omega is a domain in R-d and phi : [0, +infinity) > [0, +infinity) is a non-decreasing function satisfying some appropriate conditions. The mode of convergence is extremely delicate, and numerous problems remain open. The original motivation of our work comes from Image Processing. (C) 2016 Academie des sciences. Published by Elsevier Masson SAS.