Convolution, Rotation, and Data Fusion with Orthogonal Expansions

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Johns Hopkins University
This dissertation investigates certain special function classes, namely Hermite and Bessel functions, and uncovers some useful properties relating multiplication, convolution, rotation, and coordinate conversion. These mathematical operations are performed on the underlying basis functions and are thus continuous in nature, lending itself to higher accuracy and computational speed. Some integral transformations involving these special functions (e.g. Abel transform, 3h integrals, 3J integrals) possess recurrence relations, and so given a nite set of analytic starting conditions, higher order forms of these integrals can be obtained quickly. It will be shown how these integrals and the aforementioned special function properties are used in engineering applications including data fusion, deconvolution, continuum normal mode analysis, cryo-electron microscopy (cryo-EM) and small angle X-ray scattering (SAXS).
Integration, Hermite Functions, Bessel Functions, Recurrence Relations, Data Fusion, Rotation