DISCRETE FRAMES AND TIGHT FRAMES FOR SPARSE IMAGE REPRESENTATION

Abstract

In recent years, sparse approximation has played a fundamental role in various signal processing areas. The Gabor systems have many advantages in sparse representation. In the first part of this dissertation, we study the general theory of discrete Gabor frames by developing the Gramian and dual Gramian analysis, and construct (tight) frames using Gabor atoms to meet the needs for sparse image modeling. In the second part, we focus on constructing tight frames with Gabor and MRA structures. The experiments of image restoration illustrate the efficiency of both multi-scale Gabor induced frames and tight frames with Gabor and MRA structures in sparse image representation. In the last part, we discuss the stable and robust sparse recovery when using l1-norm as the sparsity prompting functional. Distinct requirements are imposed on the measurement matrices to ensure the stability and robustness of the models with two different loss functions.