Colloquium Jacques Morgenstern

Dans le cadre des colloquiums Jacques Morgenstern, un colloquium intitulé "Sparsity & Co.: Analysis vs Synthesis in Low-Dimensional Signal Models", par Rémy GRIBONVAL, de l'Inria, Rennes Bretagne - Atlantique.

 Résumé  :

In the past decade there has been a great interest in a synthesis-based model for signals, based on sparse and redundant representations. Such a model, which assumes that the signal of interest can be composed as a linear combination of few columns from a given matrix (the dictionary), has been extensively exploited in signal and image processing. Its applications range from compression, denoising, deblurring & deconvolution, to blind signal separation, and even more recently to new approaches to acquire and measure data with the emerging paradigm of compressive sensing.
The talk will begin with a brief review of the main existing algorithmic and theoretical results dedicated to the recovery of sparse vectors from low-dimensional projections, which form the basis of a number of signal reconstruction approaches for such generic linear inverse problems (e.g., compressed sensing, inpainting, source separation, etc.).
An alternative analysis-based model can be envisioned, where an analysis operator multiplies the signal, leading to a so-called cosparse outcome. How similar are the two signal models ? Can one derive cosparse regularization algorithms with performance guarantees when the data to be reconstructed is cosparse rather than sparse ?
Existing empirical evidence in the litterature suggests that a positive answer is likely.
In recent work we propose a uniqueness result for the solution of linear inverse problems under a cosparse hypothesis, based on properties of the analysis operator and the measurement matrix. Unlike with the synthesis model, where recovery guarantees usually require the linear independence of sets of few columns from the dictionary, our results suggest that linear dependencies between rows of the analysis operators may be desirable. The nature and potential of these new results will be discussed and illustrated with toy image processing and acoustic imaging experiments.

Informations et lieu de l'événement.