Applied Harmonic Analysis meets Compressed Sensing

One main focus of applied harmonic analysis is to develop representation systems based on particular partitions of Fourier domain aiming to provide sparse approximations for certain function classes. A well-known example are wavelet systems, which provide optimally sparse approximations of isotropic features. Since multivariate problems are however typically governed by anisotropic features in the sense of singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shock fronts in solutions of transport dominated equations, directional representation systems such as curvelets and shearlets have recently been introduced, with the novel concept of parabolic molecules providing a unified framework for sparse approximation properties of those systems.

Compressed sensing is a new research area, which was introduced in 2006, and since then has already become a key concept in various areas of applied mathematics, computer science, and electrical engineering. It surprisingly predicts that high-dimensional signals, which allow a sparse approximation by a suitable basis or, more generally, a frame, can be recovered from what was previously considered highly incomplete linear measurements by using, for instance, convex optimization or greedy type algorithms.

In this talk we will first provide an introduction to these two research areas and highlight recent developments. We will then discuss novel methodologies, which require a careful combination of those two fields, to solve problems in imaging sciences such as recovery of missing data and separation of morphologically distinct features, and present both analytic and numerical results. A discussion of future research directions will complete the talk.