Name: Gitta Kutyniok, Department of Statistics, Stanford
Title: l1 Minimization and the Geometric Separation Problem
When/where: Tuesday, May 20th, 4:15 p.m., AKW 200
Abstract:
Modern data are often composed of two (or more) geometrically 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.
In this talk we develop a theoretical approach to this Geometric Separation Problem in which a deliberately overcomplete representation is chosen made of two frames. One is suited to pointlike structures (wavelets) and the other suited to curvelike structures (curvelets or shearlets). The decomposition principle is to minimize the l_1 norm of the analysis (rather than synthesis) frame coefficients. Our theoretical results show that at all sufficiently fine scales, nearly perfect separation is indeed achieved.
Our analysis has two interesting features. Firstly, we use a viewpoint deriving from microlocal analysis to understand heuristically why separation might be possible and to organize a rigorous analysis. Secondly, we introduce some novel technical tools: cluster coherence, rather than the now-traditional singleton coherence and l_1 minimization in frame settings, including those where singleton coherence within one frame may be high.
This is joint work with David Donoho (Stanford University).