Automatic detection of sea mines in coastal regions is a difficult task due to the highly variable sea bottom conditions present in the underwater environment. Detection systems must be able to discriminate objects which vary in size, shape, and orientation from naturally occurring and man-made clutter. Additionally, these automated systems must be computationally efficient to be incorporated into unmanned underwater vehicle (UUV) sensor systems characterized by high sensor data rates and limited processing abilities. Using noncommutative group harmonic analysis, a fast, robust sea mine detection system is created. A family of unitary image transforms associated to noncommutative groups is generated and applied to side scan sonar image files supplied by Naval Surface Warfare Center Panama City (NSWC PC). These transforms project key image features, geometrically defined structures with orientations, and localized spectral information into distinct orthogonal components or feature subspaces of the image. The performance of the detection system is compared against the performance of an independent detection system in terms of probability of detection (P d ) and probability of false alarm (P fa ).

In this work we develop the family of Prometheus orthonormal sets (PONS) in the framework of certain abelian group algebras. Classical PONS, considered in 1991 by J. S. Byrnes, turned out to be a rediscovery of the 1960 construction by G. R. Welti [28], and of subsequent rediscoveries by other authors as well. This construction highlights the fundamental role played by group characters in the theory of PONS. In particular, we will relate classical PONS to idempotent systems in group algebras and show that signal expansions over classical PONS correspond to multiplications in the group algebra.The concept of a splitting sequence is critical to the construction of general PONS. We will characterize and derive closed form expressions for the collection of splitting sequences in terms of group algebra operations and group characters.The group algebras in this work are taken over direct products of the cyclic group of order 2. PONS leads to idempotent systems and ideal decompositions of these group algebras. The relationship between these special systems and ideal decompositions, and the analytic properties of PONS, is an open research topic. A second open research topic is the extension of this theory to group algebras over cyclic groups of order greater than 2.

Abelian group DSP can be completely described in terms of a special class of signals, the characters, defined by their relationship to the translations defined by abelian group multiplication. The first problem to be faced in extending classical DSP theory is to decide on what is meant by a translation. We have selected certain classes of nonabelian groups and defined translations in terms of left nonabelian group multiplications. The main distinction between abelian and nonabelian group DSP centers around the problem of character extensions. For abelian groups the solution of the character extension problem is simple. Every character of a subgroup of an abelian group A extends to a character of A. We will see that character extensions lie at the heart of several fast Fourier transform (FFT) algorithms.The nonabelian groups presented in this work will be among the simplest generalizations of abelian groups. A complete description of the DSP of an abelian by abelian semidirect product will be given. For such a group G, there exists an abelian normal subgroup A. A basis for signals indexed by G can be constructed by extending the characters of A and then by forming certain (left-) translations of these characters. The crucial point is that some of these characters of A cannot be extended to characters of G, but only to proper subgroups of G. In contrast to abelian group DSP expansions, expansions in nonabelian group DSP reflect local signal domain information over these proper subgroups. These expansions can be viewed as combined, local image-spectral image domain expansions in analogy to time-frequency expansions.This DSP leads to the definition of a certain class of group filters whose properties will be explored on simulated and recorded images. Through examples we will compare nonabelian group filters with classical abelian group filters. The main observation is that in contrast to abelian group filters, nonabelian group filters can localize in the image domain. This advantage has been used for detecting and localizing the position of geometric objects in image data sets.The works of R. Holmes M. Karpovski and E. Trachtenberg are the basis of much of the research in the application of nonabelian group

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