Imaging from Fourier Spectral Data: Problems in Discontinuity Detection, Non-harmonic Fourie r R construction and Point-spread Function Estimation by Adityavikram Viswanathan A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy ARIZONA STATE UNIVERS

Certain applications such as magnetic resonance imaging (M RI) 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 effici n y, the presence of jump discontinuities in the underlying function result in the familiar Gibbs phen omenon, with an immediate reduction in the accuracy of the method. This dissertation proposes me thods and computational tools for the efficient and accurate processing of such data, with appl ications to imaging. The relationship between local features and Fourier measurements is exploit ed to address three specific problems – the detection of jump discontinuities from Fourier data, t he 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 theconcentration edge detection method is discussed, which uses Fourier partial sums and “filter” factors known as concentration factors to a pproximate this jump information. A flexible, iterative framework is proposed for the design of t hese factors, along with the formulation of a statistical detector to detect the presence of jumps fro m n isy Fourier data. Extensions of the method to multiple dimensions as well as non-harmonic Fouri er measurements are also provided. Jump information from this method is shown to play an importa nt role in obtaining accurate reconstructions of functions from non-harmonic Fourier data. Th is is a challenging problem, typically complicated by the acquisition of spectral samples with non -uniform sampling density. The use of spectral re-projectionmethods is proposed to reduce the error caused by non-harmon ic acquisitions. These reconstructions are shown to offer great accuracy, wh ile requiring fewer input measurements than conventional Fourier methods. Results of an indirect r construction method are also provided, which uses jump information to synthesize “new” high-frequ ncy Fourier coefficients. Simulation results reveal this framework to yield highly accurate reco nstructions 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. Repres entative results are provided, demonstrating the accurate estimation of Gaussian and out-of-fo cus PSFs from blurred and noisy Fourier coefficients. iii ACKNOWLEDGEMENTS An endeavor of this magnitude usually requires several comp onents to fall in place to provide a fruitful result. While I do admit to being incredibly lucky a nd fortunate, I also feel blessed to have encountered a fine set of individuals during the course of com pleting my degree. These few words of acknowledgement hardly do justice to my sense of admirati on and gratitude to these people. I have to start with my family for providing me support and enc ouragement through times thick and thin. I hope that this degree provides some small modicum of satisfaction for the undoubtedly innumerable sacrifices they have performed over the years. T o my parents, who have spared no expense or effort to make my stay here comfortable, to my brot he – the only motivational speaker I would ever need and to my grandmother, who does not go a day wi thout a prayer in my name, I can only say thanks! I am incredibly fortunate to have been guided and tutored by a remarkable group of advisors. Professor Anne Gelb never ceases to amaze me with her incessa t nergy and enthusiasm, constant encouragement and precise guidance in matters academic and otherwise. This entire journey can be traced back to a serendipitous undertaking of one of her cl ass projects. I am still not quite sure what she saw in me, but I am certainly grateful for the goo d times, and the many people and places I have encountered through her. I am also extremely fo rtunate to count Professor Douglas Cochran as my advisor. There simply is no other person with hi s remarkably simple and effective teaching style, unique insight and judgement, and an uncann y bility to liven up a conversation. His words of advice have never failed to provide clarity, irr espective of the gravity of the situation. This work would not have been possible without the support an d guidance of Professor Rosemary Renaut. I will always be grateful for her understanding and e couragement, and determination to participate in my research, whether in person or over Skype. I would also like to thank Professor David Frakes for several fruitful discussions and conversi on , and Professors Tolga Duman and Cihan Tepedelenlioglu for insightful comments and feedbac k. I would also like to thank the personable and delightful staf f in the Departments of Mathematics & Statistics and Electrical Engineering. I can only imagine th ir pain at trudging through paperwork and records with “A. Nolastname”! And finally, to my fellow st udents and cohorts in Goldwater and elsewhere, thank you for the memories. iv TABLE OF CONTENTS Page LIST OF FIGURES . vii LIST OF TABLES . x