Clustered Sparsity and Separation of Cartoon and Texture

Abstract

Natural images are typically a composition of cartoon and texture structures. One common task is to separate such an image into two single images, one containing the cartoon part and the other containing the texture part. Recently, a powerful class of algorithms using sparse approximation and $\ell_1$ minimization has been introduced to resolve this problem, and numerous inspiring empirical results have already been obtained. In this paper we provide a theoretical study of the separation of a combination of cartoon and texture structures in a continuum model situation using this class of algorithms. The methodology we consider expands the image in a combined dictionary consisting of a curvelet frame and a Gabor frame and minimizes the $\ell_1$ norm. Sparse approximation properties then force the cartoon components into the curvelet coefficients and the texture components into the Gabor coefficients, thereby separating the image. Utilizing the fact that the coefficients are clustered geometrically, we prove that at sufficiently fine scales arbitrarily precise separation is possible. For this analysis, as a model for cartoon we consider a compactly supported function which is $C^2$ apart from a $C^2$ discontinuity curve. As a model for texture we consider locally oscillatory patterns generated by a Gabor system associated with a fixed appropriate size of the Gabor window, which is linked---satisfying an energy matching condition---to the scale of the curvelet system. In accordance with the continuum domain setting, the main ingredients of our analysis are clustered/geometric sparsity and a phase space viewpoint.