Novel Mathematical/Computational Approaches to Image Exploi

We propose to develop a unifying framework and formalism for the rapid and novel design and implementation of new families of image transforms based on the theory of groups and diffraction geometry. Our approach calls on such high-level imaging tools as graphs and aperiodic tilings to relate image data geometry with group structures on image data indexing sets and complex group algebra theory to define transforms on image data from group structures. This effort will lead to a unifying conceptual basis and algebra for structuring and developing fast algorithms for 1) traditionally used unitary transforms and their nonseparable multidimensional extensions, 2) new families of image transforms. 3) new classes of nonstandard filtering systems.Two classes of image transforms will be developed and tested. One class is based on direct sum decompositions of group algebras into left ideals and is implemented by nonabelian group filtering determined by complete systems of idempotents. The second is based on classical spectral decompositions of recoordinatized image data using aperiodic tilings and diffraction geometry. These transforms and filtering systems have clear path for digital implementation and offer greater variety, flexibility and resolution for advancing imaging technology as compared to standard methods.