Phase retrieval by iterated projections

Journal of the Optical Society of America A: Optics and Image Science, and Vision

Volumn 20, Issue 1, 2003, Pages 40-55

  Elser, Veit ^a  

^a Department of Population Medicine and Diagnostic Sciences   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CRYSTALLOGRAPHY; IMAGE RETRIEVAL; ITERATIVE METHODS; MAPPING; OBJECT RECOGNITION; PARAMETER ESTIMATION;

PHASE RETRIEVAL;

PROJECTION SYSTEMS;

EID: 0037499069     PISSN: 10847529     EISSN: None     Source Type: Journal    
DOI: 10.1364/JOSAA.20.000040     Document Type: Article

References (25)

1. R. P. Millane, "Phase retrieval in crystallography and optics, " J. Opt. Soc. Am. A 7, 394-411 (1990).
2. J. C. Dainty and 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 and Y. Yang, Vector Space Projections (Wiley, New York, 1998).
5. R. W. Gerchberg and 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).
7. A. Levi and 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).
8. H. Takajo, T. Takahashi, R. Ueda, and 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).
9. H. Takajo, T. Takahashi, and 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).
10. mod(ρ) - ρ, whereas the difference that we consider involves a projection on a priori constraints as well.
11. The term "stagnation" is interpreted to be the vanishing of the change between iterates.
12. N 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 and 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).
14. V. Elser, "Linear time heuristic for the bipartite Euclidean matching problem," (manuscript available from the author: vel0@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).
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).
20. T. Debaerdemaeker, C. Tate, and 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).
21. D. Sayre, "On least-squares refinement of the phases of crystallographic structure factors," Acta Crystallogr., Sect. A 28, 210-212(1972).
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, and R. Miller, "Crambin: a direct solution for a 400 atom structure," Acta Crystallogr., Sect. D 51, 33-38 (1995).
24. R. Miller, G. T. DeTitta, R. Jones, D. A. Langs, C. M. Weeks, and H. A. Hauptman, "On the application of the minimal principle to solve unknown structures," Science 259, 1430-1433 (1993).
25. H. H. Bauschke, P. L. Combettes, and D. R. Luke, "Phase retrieval, Gerchberg-Saxton algorithm, and Fienup variants: a view from convex optimization," J. Opt. Soc. Am. A 18, 1334-1345 (2002).