Ten years ago, Mallat and Zhang proposed the Matching Pursuit algorithm : since then, the dictionary approach to signal processing has been a very active field. In this paper, we try to give an overview of a series of recent results in the field of sparse decompositions and nonlinear approximation with redundant dictionaries. We discuss sufficient conditions on a decomposition to be the unique and simultaneous sparsest ℓr expansion for all r, 0 ≤ r ≤ 1. In particular, we prove that any decomposition has this nice property if the number of its nonzero coefficients does not exceed a quantity which we call the spread of the dictionary. After a brief discussion of the interplay between sparse decompositions and nonlinear approximation with various families of algorithms, we review several recent results that provide sufficient conditions for the Matching Pursuit, Orthonormal Matching Pursuit, and Basis Pursuit algorithms to have good recovery properties. The most general conditions are not straightforward to check, but weaker estimates based on the notions of coherence of the dictionary are recalled, and we discuss how these results can be applied to approximation and sparse compositions with highly redundant incoherent dictionaries built by taking the union of several orthonormal bases. Eventually, based on Bernstein inequalities, we discuss how much approximation power can be gained by replacing a single basis with such redundant dictionaries.
We propose a new framework, called piecewise linear separation, for blind source separation of possibly degenerate mixtures, including the extreme case of a single mixture of several sources. Its basic principle is to: 1/ decompose the observations into "components" using some sparse decomposition/nonlinear approximation technique; 2/ perform separation on each component using a "local" separation matrix. It covers many recently proposed techniques for degenerate BSS, as well as several new algorithms that we propose. We discuss two particular methods of multichannel decompositions based on the Best Basis and Matching Pursuit algorithms, as well as several methods to compute the local separation matrices (assuming the mixing matrix is known). Numerical experiments are used to compare the performance of various combinations of the decomposition and local separation methods. On the dataset used for the experiments, it seems that BB with either cosine packets of wavelet packets (Beylkin, Vaidyanathan, Battle3 or Battle 5 filter) are the best choices in terms of overall performance because they introduce a relatively low level of artefacts in the estimation of the sources; MP introduces slightly more artefacts, but can improve the rejection of the unwanted sources.