Compressed sensing signal models - to infinity and beyond?

Thomas Blumensath (1) , Michael Davis (1)

1 School of Engineering [Edinburgh]

Compressed sensing is an emerging signal acquisition technique that enables signals to be sampled well below the Nyquist rate, given a finite dimensional signal with a sparse representation in some orthonormal basis. In fact, sparsity in an orthonormal basis is only one possible signal model that allows for sampling strategies below the Nyquist rate. We discuss some recent results for more general signal models based on unions of subspaces that allow us to consider more general structured representations. These include classical sparse signal models and finite rate of innovation systems as special cases. We consider the dimensionality conditions for two aspects of the compressed sensing inverse problem: the existence of one-to-one maps to lower dimensional observation spaces and the smoothness of the inverse map. On the surface Lipschitz smoothness of the inverse map appears to limit the applicability of compressed sensing to infinite dimensional signal models. We therefore discuss conditions where smooth inverse maps are possible even in infinite dimensions. Finally we conclude by mentioning some recent work which develops the these ideas further allowing the theory to be extended beyond exact representations to structured approximations.

Domaines: Traitement du signal et de l'image [eess.SP]; Théorie de l'information et codage [math.IT]; Théorie de l'information [cs.IT]; Traitement du signal et de l'image [eess.SP]

Thomas Blumensath, Michael Davis. Compressed sensing signal models - to infinity and beyond?. SAMPTA'09, May 2009, Marseille, France. Special Session on compressed sensing. ⟨hal-00451593⟩