OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 20, Iss. 6 — Jun. 1, 2003
  • pp: 1025–1034

Hybrid projection–reflection method for phase retrieval

Heinz H. Bauschke, Patrick L. Combettes, and D. Russell Luke  »View Author Affiliations

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

View Full Text Article

Enhanced HTML    Acrobat PDF (811 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

Original Manuscript: September 20, 2002
Revised Manuscript: February 6, 2003
Manuscript Accepted: February 6, 2003
Published: June 1, 2003

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)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  2. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
  3. A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984). [CrossRef]
  4. A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 277–320.
  5. P. L. Combettes, “The foundations of set theoretic estimation,” Proc. IEEE 81, 182–208 (1993). [CrossRef]
  6. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part I—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982). [CrossRef]
  7. J. P. Boyle, R. L. Dykstra, “A method for finding projections onto the intersection of convex sets in Hilbert spaces,” Lect. Notes Stat. 37, 28–47 (1986).
  8. P.-L. Lions, B. Mercier, “Splitting algorithms for the sum of two nonlinear operators,” SIAM J. Numer. Anal. 16, 964–979 (1979). [CrossRef]
  9. H. H. Bauschke, P. L. Combettes, D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002). [CrossRef]
  10. R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990). [CrossRef]
  11. 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.
  12. D. R. Luke, J. V. Burke, R. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002). [CrossRef]
  13. H. Stark, ed., Image Recovery: Theory and Application (Academic, Orlando, Fla.1987).
  14. Typically, 0.5≤β≤1.
  15. 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.
  16. If m=0,then x=0is the unique solution of the corresponding phase-retrieval problem.
  17. For the BIO algorithm, this was already pointed out in Remark 5.6 of Ref. 9.
  18. 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.
  19. 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).
  20. 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.
  21. 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]
  22. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited