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Non-Convex Structured Phase Retrieval

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Document pages: 22 pages

Abstract: Phase retrieval (PR), also sometimes referred to as quadratic sensing, is aproblem that occurs in numerous signal and image acquisition domains rangingfrom optics, X-ray crystallography, Fourier ptychography, sub-diffractionimaging, and astronomy. In each of these domains, the physics of theacquisition system dictates that only the magnitude (intensity) of certainlinear projections of the signal or image can be measured. Without anyassumptions on the unknown signal, accurate recovery necessarily requires anover-complete set of measurements. The only way to reduce themeasurements sample complexity is to place extra assumptions on the unknownsignal image. A simple and practically valid set of assumptions is obtained byexploiting the structure inherently present in many natural signals orsequences of signals. Two commonly used structural assumptions are (i) sparsityof a given signal image or (ii) a low rank model on the matrix formed by a set,e.g., a time sequence, of signals images. Both have been explored for solvingthe PR problem in a sample-efficient fashion. This article describes this work,with a focus on non-convex approaches that come with sample complexityguarantees under simple assumptions. We also briefly describe other differenttypes of structural assumptions that have been used in recent literature.

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