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Convex Programming-based Phase Retrieval: Theory and Applications

Citation

Jaganathan, Kishore (2016) Convex Programming-based Phase Retrieval: Theory and Applications. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9C82775. http://resolver.caltech.edu/CaltechTHESIS:05312016-051759406

Abstract

Phase retrieval is the problem of recovering a signal from its Fourier magnitude. This inverse problem arises in many areas of engineering and applied physics, and has been studied for nearly a century. Due to the absence of Fourier phase, the available information is incomplete in general. Classic identifiability results state that phase retrieval of one-dimensional signals is impossible, and that phase retrieval of higher-dimensional signals is almost surely possible under mild conditions. However, there are no efficient recovery algorithms with theoretical guarantees. Classic algorithms are based on the method of alternating projections. These algorithms do not have theoretical guarantees, and have limited recovery abilities due to the issue of convergence to local optima.

Recently, there has been a renewed interest in phase retrieval due to technological advances in measurement systems and theoretical developments in structured signal recovery. In particular, it is now possible to obtain specific kinds of additional magnitude-only information about the signal, depending on the application. The premise is that, by carefully redesigning the measurement process, one could potentially overcome the issues of phase retrieval. To this end, another approach could be to impose certain kinds of prior on the signal, depending on the application. On the algorithmic side, convex programming based approaches have played a key role in modern phase retrieval, inspired by their success in provably solving several quadratic constrained problems.

In this work, we study several variants of phase retrieval using modern tools, with focus on applications like X-ray crystallography, diffraction imaging, optics, astronomy and radar. In the one-dimensional setup, we first develop conditions, which when satisfied, allow unique reconstruction. Then, we develop efficient recovery algorithms based on convex programming, and provide theoretical guarantees. The theory and algorithms we develop are independent of the dimension of the signal, and hence can be used in all the aforementioned applications. We also perform a comparative numerical study of the convex programming and the alternating projection based algorithms. Numerical simulations clearly demonstrate the superior ability of the convex programming based methods, both in terms of successful recovery in the noiseless setting and stable reconstruction in the noisy setting.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Convex optimization, Signal processing, Phase retrieval
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Electrical Engineering
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Hassibi, Babak
Thesis Committee:
  • Hassibi, Babak (chair)
  • Vaidyanathan, P. P.
  • Tropp, Joel A.
  • Chandrasekaran, Venkat
  • Yang, Changhuei
Defense Date:16 May 2016
Non-Caltech Author Email:kishorejaganathan (AT) gmail.com
Record Number:CaltechTHESIS:05312016-051759406
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:05312016-051759406
DOI:10.7907/Z9C82775
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9814
Collection:CaltechTHESIS
Deposited By: Kishore Jaganathan
Deposited On:01 Jun 2016 17:19
Last Modified:18 May 2017 17:44

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