A proximal method for inverse problems in image processing

In this paper, we present a new algorithm to solve some inverse problems coming from the field of image processing. The models we study consist in minimizing a regularizing, convex criterion under a convex and compact set. The main idea of our scheme consists in solving the underlying variational inequality with a proximal method rather than the initial convex problem. Using recent results of A. Nemirovski [13], we show that the scheme converges at least as O (1/k) (where k is the iteration counter). This is in some sense an optimal rate of convergence. Finally, we compare this approach to some others on a problem of image cartoon+texture decomposition.