A complex-valued overcomplete representation of information for visual search: A learning theoretic approach based on multiscale symmetry - Eghbalnia, Hamid R (ProQuest Dissertations & Theses, 2000)
In this thesis we develop a mathematical and computational framework for visual search as a dynamic process. Our theory and computational implementation are based on theoretical and experimental results in neuroscience pertaining to human information processing driven by visual attention. We adopt the viewpoint that a search problem is indeed driven by its goal. We coin the terminology information dynamics, referring to our formulation of search that exploits the advantages of dynamics in representation. We construct overcomplete representations relying on physical interpretation of the family of Cauchy distributions. These distributions give rise to set of directional filters with unique properties of localizability with respect to the symmetries inherent to the problem, much in the spirit of Felix Klein's Erlanger Program. The role of dynamics is to select an ordering for special symmetries (i.e. search transformations) using a set of decision operators that model the search goal. At this stage, the variational solutions pertaining to the learning-theoretic formulation for information dynamics are seen to have an interpretation in complex projective geometry. To that end, we use the Fubini-Study metric to measure the variation of features detected by the decision operators. Robust computational implementation of the theoretical steps above presents its own challenges in computational mathematics. We provide additional mathematical tools and numerical algorithms that are necessary for an effective and robust computational implementation. These methods include techniques to deal with (1) varying illumination and edge effects in digital images; (2) directional filtering; (3) robust noise enhanced extremum detection; (4) feature coding dynamics. Information dynamics is then applied to the problem of face recognition via feature discovery. As further demonstrations of the utility of the theory, we give applications based on Support Vector Machines (SVM), multiscale entropy minimization, and principal components analysis (PCA) methods. We close by considering applications of information dynamics to diverse research problems, including machine vision and feature discovery in genomics.