|Date:||Mon, November 15, 2010|
|Place:||Research II Lecture Hall|
Abstract: During the last two years, sparsity has become a key concept in various areas of mathematics, computer science, and electrical engineering. Sparsity methodologies explore the fundamental fact that many types of data/signals can be represented or at least approximated by only a few non-vanishing coefficients when choosing a suitable basis or, more generally, a frame. If signals can be sparsely approximated, they can in general be recovered from very few measurements using l1 minimization techniques, which is typically coined Compressed Sensing.
One application of this novel methodology is the separation of data, which are composed of two (or more) geometrically distinct constituents -- for instance, spines (pointlike structures) and dendrites (curvelike structures) in neurobiological imaging. Although it seems impossible to extract those components -- as there are two unknowns for every datum -- suggestive empirical results using Compressed Sensing have already been obtained.
In this talk we will first give an introduction into the concept of sparse approximation and Compressed Sensing. Then we will develop a very general theoretical approach to the problem of data separation based on these methodologies. Finally, we will apply our results to the situation of separation of pointlike and curvelike structures, for instance, in neurobiological imaging, where a deliberately overcomplete representation made of wavelets (suited to pointlike structures) and shearlets (suited to curvelike structures) will be chosen. Our theoretical results, which are based on applied harmonic analysis and microlocal analysis considerations, show that at all sufficiently fine scales, nearly-perfect separation is indeed achieved.