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Overcomplete dictionaries for sparse representation of signals

In recent years there has been a growing interest in the study of sparse representation of signals. Using an overcomplete dictionary that contains prototype signal-atoms, signals are described as linear combinations of a few of these atoms. Applications that use sparse representation are many and include compression, regularization in inverse problems, feature extraction, and more. Roughly speaking, the activity in this field concentrates around two main problems: (i) algorithms for performing sparse decomposition and their performance analysis, and (ii) dictionary composition methods. The ‘sparse decomposition’ activity deals with the pursuit problem given a signal and an overcomplete dictionary, the goal is to find the smallest set of atoms from the dictionary to represent it. As this problem is known to be NP-hard, several approximation algorithms have been suggested, such as the matching pursuit, the basis pursuit, FOCUSS, and their variants. Recent work has been also directed at the study of these algorithms’ performance and prospects of success. The construction of appropriate dictionaries that lead to the existence of sparse representations for a family of signals in mind is the second challenge addressed in recent years in this field. Among the various contributions on this problem, dictionary learning from examples is especially appealing as it can tune the found dictionary to specific and narrow families of signals. In this work we concentrate on this dictionary learning process. Starting with a wide literature survey of pursuit techniques and existing dictionary learning