Modeling oscillatory components with the homogeneous spaces ̇ BMO − α and Ẇ − α , p ∗

This paper is devoted to the decomposition of an image f into u + v, with u a piecewise-smooth or “cartoon” component, and v an oscillatory component (texture or noise), in a variational approach. The cartoon component u is modeled by a function of bounded variation, while v, usually represented by a square integrable function, is now being modeled by a more refined and weaker texture norm, as a distribution. Generalizing the idea of Y. Meyer [32], where v ∈ F = div(BMO) = ̇ BMO, we model here the texture component by the action of the Riesz potentials on v that belongs to BMO or to Lp. In an earlier work [26], the authors proposed energy minimization models to approximate (BV,F ) decompositions explicitely expressing the texture as divergence of vector fields in BMO. In this paper, we consider an equivalent more isotropic norm of the space F in terms of the Riesz potentials, and study models where the Riesz potentials of oscillatory components belong to BMO or to Lp, 1 ≤ p < ∞ (thus we consider oscillatory components in ̇ BMO or in Ẇα,p, with α < 0). Theoretical, experimental results and comparisons to validate the proposed methods are presented.