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Thesis Title

A Novel Class of Recursively Constrained Algorithms for Localized Energy Solutions : Theory and Application to Magnetoencephalography and Signal Processing

Thesis Abstract

We develop a novel approach to the recovery of are difficult to parameterize signals that have localized but complexly distributed energy. Many signal processing areas give rise to problems with such characteristics. A requirement of high resolution makes estimation of localized energy signals especially challenging. A general algorithm is developed for finding such signals. Termed FOCUSS (FOCal Underdetermined System Solution), it encompasses a class of nonparametric algorithms for high resolution estimation. The algorithm is based on recursive weighted 2-norm minimization and requires no a priori models or assumptions other than that the energy in the solution is localized. FOCUSS can be seen as a continuation of the development of adaptive algorithms for high resolution spectral estimation begun by Papoulis and Chamzas.

The mathematical motivation for the algorithm and the functional analysis theory for minimum norm based optimization are reviewed. We suggest a mathematical definition of localized energy solutions and derive conditions for uniqueness of these solutions. The proposed class of algorithms is given and proofs of local and global convergence for the algorithms and the rate of convergence are shown. Cost functions associated with different realizations of the general algorithm are derived. Ways of modifying FOCUSS to achieve particular convergence behaviors are discussed in detail. The connection of FOCUSS to existing optimization methods, including Newton's method, classical optimization, and neural networks is given.

 The initial motivation for this work comes from the problem of imaging neuroelectric activity in the cortex from magnetoencephalography (MEG) data. A detailed study of the application of FOCUSS to the MEG problem is presented.Applications to other selected problems designed to illustrate key properties of the new algorithm are presented in the last chapter. These include sinusoid detection, direction of arrival estimation, pattern classification, and a spectral estimation problem where we examine FOCUSS solutions in the case of non-localized energy signals.

Year of Graduation: 1995