High-Dimension Multilabel Problems: Convex or Nonconvex Relaxation?

This paper is concerned with the problem of relaxing non convex functionals, used in image processing, into convex problems. We review most of the recently introduced relaxation methods, and we propose a new convex one based on a probabilistic approach, which has the advantages of being intuitive, flexible and involving an algorithm without inner loops. We investigate in detail the connections between the solutions of the relaxed functionals with a minimizer of the original one. Such connection is demonstrated only for non convex relaxation which turns out to be quite robust to initialization. As a case of study, we illustrate our theoretical analysis with numerical experiments, namely for the optical flow problem.