Geometric Separation of Singularities using Combined Multiscale Dictionaries

Several empirical results that have appeared in the literature during the last decade have shown that it is often possible to separate images and other multidimensional data into geometrically distinct con-stituents. A rigorous mathematical analysis of the geometric separation problem in the two-dimensional setting was recently introduced by Donoho and Kutyniok [4], who proposed a mathematical framework to separate point and smooth curve singularities in 2D images using a combined dictionary consisting of curvelets and wavelets. In this paper, we adapt their approach and introduce a novel argument to extend geometric separation to the three-dimensional setting. We show that it is possible to separate point and piecewise linear singularities in 3D using a combined dictionary consisting of shearlets and wavelets. Our new approach takes advantage of the microlocal properties of the shearlet transform and has the ability to handle singularities containing vertices and corner points, which could not be handled using the original arguments. Key words and phrases: cluster coherence, geometric separation, ℓ1 minimization, shearlets, sparse rep-resentations, wavelets. AMS Mathematics Subject Classication: 42C15, 42C40.