This thesis addresses the problems of interferometric inversion and phase retrieval with the goal of identifying performance guarantees for deterministic imaging problems via non-convex optimization theory. Interferometry is a powerful technique that leverages correlations among diverse measurements, which is fundamentally deployed in applications for acoustic, electromagnetic, or geophysical imaging for added robustness to phase aberrations encountered in physical sensing processes. On the other hand phaseless imaging offers advantages such as reduced complexity and cost in data acquisition compared to coherently sensing the scattered fields.
We begin the thesis with presenting a non-convex optimization based approach for interferometric inversion that overcomes the limitations of conventional Fourier based techniques, and state-of-the art convex low rank matrix recovery (LRMR) approaches. To this end we take inspiration from the statistical framework of Wirtinger Flow (WF), which is a computationally efficient non-convex method for phase retrieval that relies on a sufficiently accurate initial estimate, with exact recovery guarantees applicable to probabilistic settings such as Gaussian sampling and coded diffraction patterns.
We extend the algorithmic principles of WF to accommodate the problem of interferometric inversion, and develop a novel mathematical framework in the equivalent lifted domain that bridges its theory with LRMR towards attaining recovery guarantees for deterministic models. We coin our formulation as Generalized Wirtinger Flow (GWF), and identify the theoretical and practical advantages over its convex LRMR based counterparts.
Next, we study GWF as a deterministic non-convex optimization framework. We establish the exact recovery guarantees for interferometric inversion and phase retrieval via fully geometric arguments under arbitrary measurement models. We characterize our recovery guarantees by sufficient conditions on the lifted forward map of each problem, which are ultimately less stringent than those of prominent LRMR methods in the literature. We derive the performance limits of our framework via bounds on the convergence rate and on the signal-to-noise ratio in the presence of additive noise. Additionally, we prove that interferometric inversion is a better conditioned problem than phase retrieval, where the impact of additional loss of phase information is captured in the stringency of our sufficient conditions and the subsequent performance limits for exact recovery.
We then utilize our abstract GWF framework for interferometric wave-based imaging, and establish its performance guarantees with respect to imaging parameters. We study the sufficient condition of GWF for interferometric measurements formed by deterministic wave-based imaging operators under far-field, and small scene assumptions, and determine lower bounds on the resolution and the minimal sample complexity for exact imaging. Using our bound on the resolution, we establish GWF as a super-resolution technique for imaging sufficiently small scenes. Furthermore we confirm the perpetual ill-posed nature of phaseless imaging with a deterministic design of 2D Fourier slices, where even infinitely many measurements prove insufficient for recovery guarantees under our framework.
Finally, motivated to address the ill-posed nature of the phase retrieval problem, we study the initialization component of our non-convex optimization approach. We develop a spectral estimation framework that unifies statistical foundations and heuristics pursued in the literature, and utilize Bregman divergences as similarity measures over phaseless measurements realized from arbitrary forward models. Towards designing robust initial estimates for phase retrieval, we consider well established information theoretic metrics which naturally measure distortion in strictly positive domains. Thereon, we use the universal property of Bregman representation, and propose novel spectral methods that minimize KL, and Itakura-Saito divergences under minimal distortion for generating accurate estimates for phase retrieval.;
December 2020; School of Engineering
Dept. of Electrical, Computer, and Systems Engineering;
Rensselaer Polytechnic Institute, Troy, NY
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