Versatile and scalable cosparse methods for physics-driven inverse problems
Résumé
Solving an underdetermined inverse problem implies the use of a regularization hypothesis. Among possible regularizers, the so-called sparsity hypothesis, described as a synthesis model of the signal of interest from a low number of elementary signals taken in a dictionary, is now widely used. In many inverse problems of this kind, it happens that an alternative model, the cosparsity hypothesis (stating that the result of some linear analysis of the signal is sparse), offers advantageous properties over the classical synthesis model. A particular advantage is its ability to intrinsically integrate physical knowledge about the observed phenomenon, which arises naturally in the remote sensing contexts through some underlying partial differential equation. In this chapter, we illustrate on two running examples (acoustic source localization and brain source imaging) the power of a generic cosparse approach to a wide range of problems governed by physical laws, how it can be adapted to each of these problems in a very versatile fashion, and how it can scale up to large volumes of data typically arising in applications.
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