Rank related properties, for Basis Pursuit and total variation regularization
Résumé
This paper focuses on optimization problems containing an $l^1$ kind of regularity criterion and a smooth data fidelity term. A general theorem is adapted to this context. It gives an estimate of the distribution law of the "rank" of the solution to the optimization problems, when the initial datum follows a uniform (in a convex compact set) distribution law. It says that asymptotically, solution with a large rank are more and more likely. The main goal of this paper is to understand the meaning of this notion of rank for some energies which are commonly used in image processing. We study in details the energy whose level sets are defined as the convex hull of finite subset of $\RR^N$ (think Basis Pursuit) and the total variation. For these energies, the notion of rank respectively relates to sparse representation and staircasing.