A note on sparse reconstruction methods

In this paper we discuss some aspects of sparse reconstruction techniques for inverse problems, which recently became popular due to several superior properties compared to linear reconstructions. We briefly review the standard sparse reconstructions based on l1-minimization of coefficients with respect to an orthonormal basis, and also some recently proposed improvements based on Bregman iterations and inverse scale space techniques. For the latter we provide uniqueness results not available before for inverse problems with sparsity constraints. In order to gain further understanding of sparse reconstruction techniques we provide a detailed analysis in the singular basis for the operator describing the inverse problem. This allows to compute analytic expressions for the reconstructions and highlight certain features. We also show that a very classical linear reconstruction technique, the truncated singular value decomposition is indeed equivalent to a sparse reconstruction technique with data-dependent weights. Finally we touch the question whether it pays off to use sparse reconstruction schemes directly for the full inverse problem or if simple two-step schemes, consisting of a linear inversion and subsequent shrinkage, can potentially yield results of similar quality