## Hybrid projection–reflection method for phase retrieval

JOSA A, Vol. 20, Issue 6, pp. 1025-1034 (2003)

http://dx.doi.org/10.1364/JOSAA.20.001025

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### Abstract

The phase-retrieval problem, fundamental in applied physics and engineering, addresses the question of how to determine the phase of a complex-valued function from modulus data and additional *a priori* information. Recently we identified two important methods for phase retrieval, namely, Fienup’s basic input–output and hybrid input–output (HIO) algorithms, with classical convex projection methods and suggested that further connections between convex optimization and phase retrieval should be explored. Following up on this work, we introduce a new projection-based method, termed the hybrid projection–reflection (HPR) algorithm, for solving phase-retrieval problems featuring nonnegativity constraints in the object domain. Motivated by properties of the HPR algorithm for convex constraints, we recommend an error measure studied by Fienup more than 20 years ago. This error measure, which has received little attention in the literature, lends itself to an easily implementable stopping criterion. In numerical experiments we found the HPR algorithm to be a competitive alternative to the HIO algorithm and the stopping criterion to be reliable and robust.

© 2003 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 20, 2002

Revised Manuscript: February 6, 2003

Manuscript Accepted: February 6, 2003

Published: June 1, 2003

**Citation**

Heinz H. Bauschke, Patrick L. Combettes, and D. Russell Luke, "Hybrid projection–reflection method for phase retrieval," J. Opt. Soc. Am. A **20**, 1025-1034 (2003)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-20-6-1025

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### References

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- In general, the object-domain space need not be restricted to real-valued signals. For a review of a phase-retrieval application in which the iterates are complex valued, see Ref. 12.
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- Typically, 0.5≤β≤1.
- It was pointed out in Remark 4.1 of Ref. 9that a more literal reformulation of Eq. (13) leads to (31)(∀t∈X)xn+1(t)=(PM(xn))(t)if t∈D or (PM(xn))(t)=0xn(t)-β(PM(xn))(t)otherwise.Under the assumption that the zero crossings of PM(xn)outside of Dare negligible, Eq. (31) reduces to Eq. (14). In digital computing, this assumption is justified by the fact that the probability of obtaining zero numbers is virtually zero.
- If m=0,then x=0is the unique solution of the corresponding phase-retrieval problem.
- For the BIO algorithm, this was already pointed out in Remark 5.6 of Ref. 9.
- We stress that monitoring the sequences (PM(xn))n∈Nand (PS+PM(xn))n∈Nis well motivated from the convex consistent setting. Replace the nonconvex set Mand its corresponding projector PMwith the convex set Band the corre-sponding projector PB.If S+∩B≠∅,then, using the results in Ref. 9, one can prove that ES+(xn)→0with equality precisely when PB(xn)∈S+∩B.However, if S+∩B=∅,which is likely a better approximation of the geometry of the phase-retrieval problem, then minimizing ES+(xn)corresponds to finding a displacement vector for S+and Bin the sense of Ref. 19. If the problem is feasible but ES+(xn)is positive, then the algorithm has stagnated, and the value of ES+(xn)is an indicator of the quality of the stagnation point.
- H. H. Bauschke, J. M. Borwein, “On the convergence of von Neumann’s alternating projection algorithm for two sets,” Set-Valued Anal. 1, 185–212 (1993).
- J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 231–275.
- T. R. Crimmins, J. R. Fienup, B. J. Thelen, “Improved bound on object support from autocorrelation support and application to phase retrival,” J. Opt. Soc. Am. A 7, 3–13 (1990). [CrossRef]
- J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986). [CrossRef]

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