Algebraic Signal Processing Theory: Towards Multiresolution Analysis

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Signal processing is the enabler and driving force ("brain in the box") behind many everyday technologies including audio/image/video processing (MP3, JPEG, MPEG), communications (cell phones), medical and bioimaging (MRI, fMRI, CAT/PET scan, high-throughput drug screening), sensor networks (cars, surveillance), security (biometrics for financial industry and securing US borders). As a discipline, signal processing makes heavy use of deep mathematics including complex calculus, linear algebra, stochastics and approximation theory. In fact, many of the above advances were made possible by importing and using novel techniques from mathematics. This research connects signal processing to abstract algebra (a major discipline in mathematics) at a fundamental level. In doing so, a whole new set of mathematical tools becomes available to signal processing. In preliminary work, these tools have already produced novel andefficient processing techniques. The investigators will develop many more and aim to tackle problems that have eluded solution with previous methods.The platform for the research is a novel approach to signal processing, called algebraic signal processing theory (ASP), which will be further developed in this work. ASP generalizes the standard linear signal processing to provide novel notions of filtering, Fourier transforms, and others. ASP captures many existing signal processing methods into one common framework and enables the derivation of new ones in one and higher dimensions, separable and nonseparable, shift-invariant and shift-variant. A major goal in this research is to expand ASP to include general notions of filter banks and ultiresolution methods for important applications. At a fundamental level, ASP may provide a more rigorous, axiomatic approach to signal processing and thus impact education. A first course based on ASP will be developed in this project.