Analysis of lscr1 minimization in the Geometric Separation Problem

Modern data are often composed of two (or more) morphologically distinct constituents - for instance, pointlike and curvelike structures in astronomical imaging of galaxies. Although it seems impossible to extract those components - as there are two unknowns for every datum - suggestive empirical results have already been obtained especially by Jean-Luc Starck and collaborators. In this paper we develop a theoretical view-point, defining a Geometric Separation Problem and analyzing a model procedure. This procedure is inspired by work relating lscr1 minimization and sparsity. The procedure uses two deliberately overcomplete systems which sparsify the different components and decomposes by lscr1 minimization of the analysis (rather than synthesis) frame coefficients. We formalize two concepts - cluster coherence in place of the now-traditional singleton coherence and lscr1 minimization in frame settings, including those where singleton coherence within one frame may be high - and develop all the needed machinery to make these into fruitful tools. Our general approach applies to the problem of geometric separation of pointlike and curvelike structures in images by employing frames of radial wavelets and curvelets or orthonormal wavelets and shearlets. Our theoretical results show that at all sufficiently fine scales, nearly-perfect separation is achieved. We use microlocal analysis to understand heuristically why separation might be possible and to organize a rigorous analysis.