The Analysis Sparse Model - Definition, Pursuit, Dictionary Learning, and Beyond
January 16th, 2012
Mathematics and Image Analysis 2012 Workshop (MIA'12), Paris

The synthesis-based sparse representation model for signals has drawn a considerable interest in the past decade. Such a model assumes that the signal of interest can be decomposed as a linear combination of a few atoms from a given dictionary. In this talk we concentrate on an alternative, analysis-based model, where an analysis operator — hereafter referred to as the “Analysis Dictionary” – multiplies the signal, leading to a sparse outcome. While the two alternative models seem to be very close and similar, they are in fact very different. In this talk we define clearly the analysis model and describe how to generate signals from it. We discuss the pursuit denoising problem that seeks the zeros of the signal with respect to the analysis dictionary given noisy measurements. Finally, we explore ideas for learning the analysis dictionary from a set of signal examples. We demonstrate this model’s effectiveness in several experiments, treating synthetic data and real images, showing a successful and meaningful recovery of the analysis dictionary.

Invited talk. This talk was also given as keynote talk in LVA-ICA, March 13th, 2012, Tel-Aviv, and in Oberwolfach (Germany) workshop on harmonic analysis on June 13th, 2012. Joint work with Ron Rubinstein (former PhD student), Tomer peleg (PhD student), Remi Gribonval and Sangnam Nam (INRIA, Rennes), and Mike Davies (UEdin).