This manuscript is a synthesis of my research activity at CRAN between 2005 and 2013. My research projects deal with inverse problems in signal and image processing, sparse approximation, hyperspectral image analysis, and 3D image reconstruction. I pay specific attention to the development, analysis and utilization of sparse approximation algorithms for inverse problems characterized by ill-conditioned dictionaries. In the first chapter, heuristic algorithms are proposed to minimize mixed L2-L0 cost functions. These are ''bidirectional'' greedy algorithms defined as extensions of Orthogonal Least Squares (OLS). Indeed, empirical comparisons show that OLS and its derived versions behave nicely when the dictionary is an ill-conditioned matrix. The second chapter is an applicative part in atomic force microscopy, where the OLS based algorithms are utilized with a specific dictionary in order to perform automatic segmentation of signals. This segmentation leads to the reconstruction of a set of 2D images representing electrostatic and bio-mechanical properties at the nanoscale. The third chapter is a theoretical study aiming to analyze the greedy algorithms OMP (Orthogonal Matching Pursuit) and OLS. A first k-step recovery analysis or OLS is provided. Then, the worst case exact recovery conditions are being thoroughly evaluated for both OMP and OLS when a number of iterations have already been performed. The comparisons validate the better behavior of OLS for problems involving ill-conditioned dictionaries. The fourth chapter sketches a few perspectives, both methodological and applicative, regarding sparse analysis for inverse problems.
