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>Imaging from Fourier Spectral Data: Problems in Discontinuity Detection, Non-harmonic Fourier Reconstruction and Point-spread Function Estimation.
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Imaging from Fourier Spectral Data: Problems in Discontinuity Detection, Non-harmonic Fourier Reconstruction and Point-spread Function Estimation.
Certain applications such as magnetic resonance imaging (MRI) and synthetic aperture radar (SAR) imaging demand the processing of input data collected in the spectral domain. While spectral methods traditionally boast of superior accuracy and efficiency, the presence of jump discontinuities in the underlying function result in the familiar Gibbs phenomenon, with an immediate reduction in the accuracy of the method. This dissertation proposes methods and computational tools for the efficient and accurate processing of such data, with applications to imaging. The relationship between local features and Fourier measurements is exploited to address three specific problems -- the detection of jump discontinuities from Fourier data, the reconstruction of functions from non-harmonic or non-uniform Fourier measurements, and the estimation of point-spread functions (PSFs) from blurred Fourier data.;Jump locations and values are among the most important local features of a piecewise-analytic function. Use of the concentration edge detection method is discussed, which uses Fourier partial sums and "filter" factors known as concentration factors to approximate this jump information. A flexible, iterative framework is proposed for the design of these factors, along with the formulation of a statistical detector to detect the presence of jumps from noisy Fourier data. Extensions of the method to multiple dimensions as well as non-harmonic Fourier measurements are also provided. Jump information from this method is shown to play an important role in obtaining accurate reconstructions of functions from non-harmonic Fourier data. This is a challenging problem, typically complicated by the acquisition of spectral samples with non-uniform sampling density. The use of spectral re-projection methods is proposed to reduce the error caused by non-harmonic acquisitions. These reconstructions are shown to offer great accuracy, while requiring fewer input measurements than conventional Fourier methods. Results of an indirect reconstruction method are also provided, which uses jump information to synthesize "new" high-frequency Fourier coefficients. Simulation results reveal this framework to yield highly accurate reconstructions of the underlying function. Finally, the presence of jump discontinuities in a function is exploited to construct an efficient scheme for the estimation of PSFs from blurred Fourier data. Representative results are provided, demonstrating the accurate estimation of Gaussian and out-of-focus PSFs from blurred and noisy Fourier coefficients.
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