Phase retrieval, Gerchberg-Saxton algorithm, and Fienup variants: A view from convex optimization

Bauschke, Heinz H. and Combettes, Patrick L. and Luke, D. Russell (2001) Phase retrieval, Gerchberg-Saxton algorithm, and Fienup variants: A view from convex optimization. [Preprint]

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      The phase retrieval problem is of paramount importance in various areas of applied physics and engineering. The state of the art for solving this problem in two dimensions relies heavily on the pioneering work of Gerchberg, Saxton, and Fienup. Despite the widespread use of the algorithms proposed by these three researchers, current mathematical theory cannot explain their remarkable success. Nevertheless, great insight can be gained into the behavior, the shortcomings, and the performance of these algorithms from their possible counterparts in convex optimization theory. An important step in this direction was made two decades ago when the Gerchberg-Saxton algorithm was identified as a nonconvex alternating projection algorithm. The purpose of this paper is to formulate the phase retrieval problem with mathematical care and to establish new connections between well established numerical phase retrieval schemes and classical convex optimization methods. Specifically, it is shown that Fienup's basic input-output algorithm corresponds to Dykstra's algorithm, and that Fienup's hybrid input-output algorithm can be viewed as an instance of the Douglas-Rachford algorithm. This work provides a theoretical framework to better understand and, potentially, improve existing phase recovery algorithms.

      Item Type: Preprint
      Additional Information: pubdom FALSE
      Subjects: 49-xx Calculus of variations and optimal control; optimization > 49Nxx Miscellaneous topics
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 24 Nov 2003
      Last Modified: 21 Apr 2010 11:13

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