## Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization

JOSA A, Vol. 19, Issue 7, pp. 1334-1345 (2002)

http://dx.doi.org/10.1364/JOSAA.19.001334

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### Abstract

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 error reduction algorithm was identified as a nonconvex alternating projection algorithm. Our purpose 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. We provide a theoretical framework to better understand and, potentially, to improve existing phase recovery algorithms.

© 2002 Optical Society of America

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(100.3020) Image processing : Image reconstruction-restoration

(100.5070) Image processing : Phase retrieval

**History**

Original Manuscript: September 25, 2001

Revised Manuscript: January 15, 2002

Manuscript Accepted: January 15, 2002

Published: July 1, 2002

**Citation**

Heinz H. Bauschke, Patrick L. Combettes, and D. Russell Luke, "Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization," J. Opt. Soc. Am. A **19**, 1334-1345 (2002)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-19-7-1334

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- While PBis nonexpansive and therefore Lipschitz continuous, this property is not sufficient to draw the conclusion advertised in Corollary 6.1 in Ref. 88, namely (in our context), that (PBxn)converges weakly to a point in A∩B.Such a conclusion requires additional assumptions, e.g., that PBbe weakly continuous (if so, then PBxn⇀PBx), as is the case when dim H<+∞(or when Bis a closed affine subspace). Note, however, that the projector onto a closed convex set may fail to be weakly continuous. An example is on page 245 in Ref. 109.

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