Geometric Separation using a Wavelet-Shearlet Dictionary

Astronomical images of galaxies can be modeled as a superposition of pointlike and curvelike structures. Astronomers typically face the problem of extracting those components as accurate as possible. Although this problem seems unsolvable – as there are two unknowns for every datum – suggestive empirical results have been achieved by employing a dictionary consisting of wavelets and curvelets combined with `1 minimization techniques. In this paper we present a theoretical analysis in a model problem showing that accurate geometric separation can be achieved by `1 minimization. We introduce the notions of cluster coherence and clustered sparse objects as a machinery to show that the underdetermined system of equations can be stably solved by `1 minimization. We prove that not only a radial wavelet-curvelet dictionary achieves nearly-perfect separation at all sufficiently fine scales, but, in particular, also an orthonormal wavelet-shearlet dictionary, thereby proposing this dictionary as an interesting alternative for geometric separation of pointlike and curvelike structures. To derive this final result we show that curvelets and shearlets are sparsity equivalent in the sense of a finite p-norm (0 < p ≤ 1) of the cross-Grammian matrix.