During the last two years, sparsity has become a key concept in various areas
of applied mathematics, computer science, and electrical engineering. Sparsity
methodologies explore the fundamental fact that many types of data/signals can
be represented by only a few non-vanishing coefficients when choosing a suitable
basis or, more generally, a frame. If signals possess such a sparse representation,
they can in general be recovered from few measurements using $\ell_1$ minimization
techniques.
One application of this novel methodology is the geometric separation of data,
which is 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 using sparsity
considerations have already been obtained.
In this talk we will first give an introduction into the concept of sparse
representations and sparse recovery. Then we will develop a very general
theoretical approach to the problem of geometric separation based on these
methodologies by introducing novel ideas such as geometric clustering of
coefficients. Finally, we will apply our results to the situation of separation
of pointlike and curvelike structures in astronomical imaging of galaxies,
where a deliberately overcomplete representation made of wavelets (suited
to pointlike structures) and curvelets/shearlets (suited to curvelike
structures) will be chosen. The decomposition principle is to minimize the
$\ell_1$ norm of the frame coefficients. Our theoretical results, which
are based on microlocal analysis considerations, show that at all sufficiently
fine scales, nearly-perfect separation is indeed achieved.
This is joint work with David Donoho (Stanford University).