Sparsity Driven Solutions to Linear and Quadratic Inverse Problems

Abstract

The problem of signal reconstruction from inaccurate and possibly incomplete set of linear/non-linear measurements occurs in a variety of signal and image processing applications. In this thesis, we develop reconstruction algorithms that exploit signal sparsity in such settings. The assumption of sparsity is practically relevant, since most signals encountered in real-world applications admit a sparse representation in an appropriately chosen bases. We consider two measurement models for signal acquisition, namely linear and quadratic. Two inverse problems are considered under the linear model, namely dictionary learning and sparse coding, corresponding to the cases when the forward linear operator is unknown and known, respectively. The quadratic measurement model considered in our work arises in the so-called phase retrieval problem encountered in several imaging applications