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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 20, Iss. 1 — Jan. 1, 2003
  • pp: 40–55

Phase retrieval by iterated projections

Veit Elser  »View Author Affiliations

JOSA A, Vol. 20, Issue 1, pp. 40-55 (2003)

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Several strategies in phase retrieval are unified by an iterative “difference map” constructed from a pair of elementary projections and three real parameters. For the standard application in optics, where the two projections implement Fourier modulus and object support constraints, respectively, the difference map reproduces the “hybrid” form of Fienup’s input–output map when a particular choice is made for two of the parameters. The geometric construction of the difference map illuminates the distinction between its fixed points and the recovered object, as well as the mechanism whereby the form of stagnation encountered by alternating projection schemes is avoided. When support constraints are replaced by object histogram or atomicity constraints, the difference map lends itself to crystallographic phase retrieval. Numerical experiments with synthetic data suggest that structures with hundreds of atoms can be solved.

© 2003 Optical Society of America

OCIS Codes
(100.5070) Image processing : Phase retrieval

Original Manuscript: March 22, 2002
Revised Manuscript: May 1, 2002
Manuscript Accepted: August 1, 2002
Published: January 1, 2003

Veit Elser, "Phase retrieval by iterated projections," J. Opt. Soc. Am. A 20, 40-55 (2003)

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  1. R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990). [CrossRef]
  2. 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), Chap. 7, pp. 231–275.
  3. C. Giacovazzo, Direct Phasing in Crystallography (Oxford U. Press, Oxford, UK, 1998).
  4. H. Stark, Y. Yang, Vector Space Projections (Wiley, New York, 1998).
  5. 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).
  6. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
  7. 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]
  8. H. Takajo, T. Takahashi, R. Ueda, M. Taninaka, “Study on the convergence property of the hybrid input output algorithm used for phase retrieval,” J. Opt. Soc. Am. A 15, 2849–2861 (1998). [CrossRef]
  9. H. Takajo, T. Takahashi, T. Shizuma, “Further study on the convergence property of the hybrid input output algorithm used for phase retrieval,” J. Opt. Soc. Am. A 16, 2163–2168 (1999). [CrossRef]
  10. The “difference map” considered here should not be confused with the “map” of the electron density studied by crystallographers in “difference Fourier synthesis.” The latter, in our notation, corresponds to πmod(ρ)-ρ, whereas the difference that we consider involves a projection on a prioriconstraints as well.
  11. The term “stagnation” is interpreted to be the vanishing of the change between iterates.
  12. The term “subspace” denotes a general subset of EN and in almost all cases of interest is a smooth submanifold, possibly with boundary. When encountered in the discussion, linear or affine subspaces will be explicitly identified as such.
  13. K. Y. J. Zhang, P. Main, “Histogram matching as a new density modification technique for phase refinement and extension of protein molecules,” Acta Crystallogr. Sect. A 46, 41–46 (1990). [CrossRef]
  14. V. Elser, “Linear time heuristic for the bipartite Euclidean matching problem,” (manuscript available from the author: ve10@cornell.edu).
  15. In the case of support constraint with positivity, the constraint subspace is a smooth space with boundary, and the relevant dimensionality is that of the space without boundary.
  16. A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963). [CrossRef]
  17. For a smooth subspace C with tangent space X at a∈C, the affine approximation to C at a is the space C′=X+a. By taking the set of all differences of elements in C′, one recovers the linear space: X=C′-C′.
  18. V. Elser, “Random projections and the optimization of an algorithm for phase retrieval,” J. Phys. A. Math. Gen. 35, 1–13 (2002).
  19. D. Sayre, “The squaring method: a new method for phase determination,” Acta Crystallogr. 5, 60–65 (1952). [CrossRef]
  20. T. Debaerdemaeker, C. Tate, M. M. Woolfson, “On the application of phase relationships to complex structures. XXVI. Developments of the Sayre-equation tangent formula,” Acta Crystallogr., Sect. A 44, 353–357 (1988). [CrossRef]
  21. D. Sayre, “On least-squares refinement of the phases of crystallographic structure factors,” Acta Crystallogr., Sect. A 28, 210–212 (1972). [CrossRef]
  22. H. A. David, Order Statistics, 2nd ed. (Wiley, New York, 1981).
  23. C. M. Weeks, H. A. Hauptman, G. D. Smith, R. H. Blessing, M. M. Teeter, R. Miller, “Crambin: a direct solution for a 400 atom structure,” Acta Crystallogr., Sect. D 51, 33–38 (1995). [CrossRef]
  24. R. Miller, G. T. DeTitta, R. Jones, D. A. Langs, C. M. Weeks, H. A. Hauptman, “On the application of the minimal principle to solve unknown structures,” Science 259, 1430–1433 (1993). [CrossRef] [PubMed]
  25. H. H. Bauschke, P. L. Combettes, D. R. Luke, “Phase retrieval, Gerchberg–Saxton algorithm, and Fienup vari ants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002). [CrossRef]

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