Sparse Approximation and Atomic Decomposition: Considering Atom Interactions in Evaluating and Building Signal Representations

Abstract

This dissertation makes contributions to the sparse approximation and efficient representation of complex signals, e.g., acoustic signals, using greedy iterative descent pursuits and overcomplete dictionaries. As others have noted before, peculiar problems arise when a signal model is mismatched to the signal content, and a pursuit makes bad selections from the dictionary. These result in models that contain several atoms having no physical signicance to the signal, and instead exist to correct the representation through destructive interference. These "spurious" terms greatly diminish the efficiency of the generated signal model, and hinder the useful application of sparse approximation to signal analysis (e.g., source identication), visualization (e.g., source selection), and modication (e.g., source extraction). While past works have addressed these problems by reformulating a pursuit to avoid them, such as adding restrictions to the content of a dictionary, in this dissertation we use these corrective terms to learn about the signal, the pursuit algorithm, the dictionary, and the created model. Our thesis is essentially that a better signal model results when a pursuit builds it considering the interaction between the atoms. We formally study these effects and propose novel measures of them to quantify the interaction between atoms in a model, and to illuminate the role of each atom in representing a signal. We then propose and study three dierent ways of incorporating these new measures into the atom selection criteria of greedy iterative descent pursuits, and show analytically and empirically that these interference-adaptive pursuits can produce models with increased efficiency and meaningfulness, e.g., the direct correspondence between an atom and specic signal content. Finally, we propose creating a higher-level model of the decomposed signal by agglomerating the atoms of a representation into molecules based on a set of similarity criteria, and compare this method with a previous pursuit that builds molecules simultaneously with the decomposition process. In both cases, we find that the resulting molecules have a more clear relationship with the signal content.