Project B6: Sparse adaptive multiscale representations of signals and images based on advanced exponential analysis methods

In recent years, there has been significant interest in exponential analysis methods, in part, due to successful applications in recovery problems in which an unknown signal is known to possess a predetermined structure.

This project investigates connections between numerical schemes for exponential analysis (finite rate of innovation or Prony methods) and non-convex optimization algorithms, in the context of sparse approximation of signals within a large (but finite) dictionary.

Further, it aims to establish a new adaptive wavelet-like transform that permits arbitrary real translations at each scaling level. For approximation of a given function, suitable “active” translates are obtained adaptively via an exponential analysis approach. In 2D, directionality is ensured using window functions with restricted support in the frequency domain. Decomposition algorithms of this type are of high interest in signal analysis and sparse representation of images as arises, for instance, in MRI.

Methods: structured matrix decompositions, non-convex optimization, Prony’s method, non-linear approximation, multiresolution, greedy algorithms
Applications: sparse data representation, analysis of non-stationary signals