Abstract

We report computer observations on the performance of an improved version of a simulated-annealing algorithm that was used before for the problem of phase retrieval. According to the results, we propose to use this method in conjunction with the algorithm of Fienup [ Opt. Lett. 3, 27 ( 1978); Opt. Eng. 18529, ( 1979); Appl. Opt. 21, 2758 ( 1982). The full power of the simulated-annealing algorithm with large arrays appears to be limited by present-day computers rather than by its numerical performance, but we believe that this combination may constitute an efficient method for phase retrieval.

© 1988 Optical Society of America

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References

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  1. Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  2. M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
    [CrossRef]
  3. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  4. J. R. Fienup, “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
    [CrossRef]
  5. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  6. C. Y. C. Liu, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
    [CrossRef]
  7. P. J. Napier, R. H. T. Bates, “Inferring phase information from modulus information in two-dimensional aperture synthesis,” Astron. Astrophys. Suppl. 15, 427–430 (1974).
  8. B. R. Frieden, D. G. Currie, “On unfolding the autocorrelation function,” J. Opt. Soc. Am. 66, 1111 (A) (1976).
  9. R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik 61, 247–262 (1982);K. L. Garden, R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. II: One-dimensional considerations,” Optik 62, 131–142 (1982);W. R. Fright, R. M. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. III: Computational examples for two dimensions,” Optik 62, 219–230 (1982).
  10. H. H. Arsenault, K. Chalasinska-Macukov, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983);K. Chalasinska-Macukov, H. Arsenault, “Fast iterative solution to exact equations for the two-dimensional phase problem,” J. Opt. Soc. Am. A 2, 46–50 (1985).
    [CrossRef]
  11. 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]
  12. P. L. Van Hove, M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1286–1293 (1983).
    [CrossRef]
  13. H. V. Deighton, M. S. Scivier, M. A. Fiddy, “Solution of the two-dimensional phase problem,” Opt. Lett. 10, 250–251 (1985).
    [CrossRef] [PubMed]
  14. M. Nieto-Vesperinas, “A study on the performance of least square optimization methods in the problem of phase retrieval,” Opt. Acta 33, 713–722 (1986).
    [CrossRef]
  15. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  16. J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.
  17. S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
    [CrossRef] [PubMed]
  18. N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
    [CrossRef]
  19. W. E. Smith, H. H. Barrett, R. G. Paxman, “Reconstruction of objects from coded images by simulated annealing,” Opt. Lett. 8, 199–201 (1983);W. E. Smith, R. G. Paxman, H. H. Barrett, “Image reconstruction from coded data: I. Reconstruction algorithms and experimental results,” J. Opt Soc. Am. A 2, 491–500 (1985).
    [CrossRef] [PubMed]
  20. H. H. Barrett, W. E. Smith, R. G. Paxman, “Monte Carlo methods in optics,” Acta Polytech. Scand. Appl. Phys. Ser. 149, 3 (1985).
  21. M. Nieto-Vesperinas, J. A. Méndez, “Phase retrieval by Monte Carlo methods,” Opt. Commun. 59, 249–254 (1986).
    [CrossRef]
  22. M. Pincus, “A closed form solution of certain programming problems,” Oper. Res. 16, 690–694 (1968);“A Monte Carlo method for the approximate solution of certain types of constrained optimization problems,” Oper. Res. 18, 1225–1228 (1970).
    [CrossRef]
  23. J. M. Hammersley, D. C. Handscomb, Monte Carlo Methods (Methuen, London, 1967), p. 113.
  24. R. Y. Rubinstein, Simulation and the Monte Carlo Methods (Wiley, New York, 1981), p. 260.
  25. F. J. Fuentes, M. Nieto-Vesperinas, R. Navarro, “A fortan program to obtain a function of two variables from its autocorrelation,” submitted to Trans. Math. Software.
  26. A. H. Greenaway, J. G. Walker, J. A. G. Coombs, “Ambiguities in speckle reconstructions. Some ways of avoiding them,” in Indirect Imaging, J. A. Roberts, ed. (Cambridge U. Press, Cambridge, 1984), pp. 111–117.
  27. F. J. Fuentes, Instituto de Astrofísica de Canarias, La Laguna Tenerife, Spain (personal communication).

1986

M. Nieto-Vesperinas, “A study on the performance of least square optimization methods in the problem of phase retrieval,” Opt. Acta 33, 713–722 (1986).
[CrossRef]

J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
[CrossRef]

M. Nieto-Vesperinas, J. A. Méndez, “Phase retrieval by Monte Carlo methods,” Opt. Commun. 59, 249–254 (1986).
[CrossRef]

1985

H. V. Deighton, M. S. Scivier, M. A. Fiddy, “Solution of the two-dimensional phase problem,” Opt. Lett. 10, 250–251 (1985).
[CrossRef] [PubMed]

H. H. Barrett, W. E. Smith, R. G. Paxman, “Monte Carlo methods in optics,” Acta Polytech. Scand. Appl. Phys. Ser. 149, 3 (1985).

1984

1983

P. L. Van Hove, M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1286–1293 (1983).
[CrossRef]

H. H. Arsenault, K. Chalasinska-Macukov, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983);K. Chalasinska-Macukov, H. Arsenault, “Fast iterative solution to exact equations for the two-dimensional phase problem,” J. Opt. Soc. Am. A 2, 46–50 (1985).
[CrossRef]

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

W. E. Smith, H. H. Barrett, R. G. Paxman, “Reconstruction of objects from coded images by simulated annealing,” Opt. Lett. 8, 199–201 (1983);W. E. Smith, R. G. Paxman, H. H. Barrett, “Image reconstruction from coded data: I. Reconstruction algorithms and experimental results,” J. Opt Soc. Am. A 2, 491–500 (1985).
[CrossRef] [PubMed]

1982

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik 61, 247–262 (1982);K. L. Garden, R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. II: One-dimensional considerations,” Optik 62, 131–142 (1982);W. R. Fright, R. M. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. III: Computational examples for two dimensions,” Optik 62, 219–230 (1982).

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

1979

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

J. R. Fienup, “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
[CrossRef]

1978

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[CrossRef] [PubMed]

1976

B. R. Frieden, D. G. Currie, “On unfolding the autocorrelation function,” J. Opt. Soc. Am. 66, 1111 (A) (1976).

1974

P. J. Napier, R. H. T. Bates, “Inferring phase information from modulus information in two-dimensional aperture synthesis,” Astron. Astrophys. Suppl. 15, 427–430 (1974).

1973

C. Y. C. Liu, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

1968

M. Pincus, “A closed form solution of certain programming problems,” Oper. Res. 16, 690–694 (1968);“A Monte Carlo method for the approximate solution of certain types of constrained optimization problems,” Oper. Res. 18, 1225–1228 (1970).
[CrossRef]

1953

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Arsenault, H. H.

H. H. Arsenault, K. Chalasinska-Macukov, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983);K. Chalasinska-Macukov, H. Arsenault, “Fast iterative solution to exact equations for the two-dimensional phase problem,” J. Opt. Soc. Am. A 2, 46–50 (1985).
[CrossRef]

Barrett, H. H.

H. H. Barrett, W. E. Smith, R. G. Paxman, “Monte Carlo methods in optics,” Acta Polytech. Scand. Appl. Phys. Ser. 149, 3 (1985).

W. E. Smith, H. H. Barrett, R. G. Paxman, “Reconstruction of objects from coded images by simulated annealing,” Opt. Lett. 8, 199–201 (1983);W. E. Smith, R. G. Paxman, H. H. Barrett, “Image reconstruction from coded data: I. Reconstruction algorithms and experimental results,” J. Opt Soc. Am. A 2, 491–500 (1985).
[CrossRef] [PubMed]

Bates, R. H. T.

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik 61, 247–262 (1982);K. L. Garden, R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. II: One-dimensional considerations,” Optik 62, 131–142 (1982);W. R. Fright, R. M. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. III: Computational examples for two dimensions,” Optik 62, 219–230 (1982).

P. J. Napier, R. H. T. Bates, “Inferring phase information from modulus information in two-dimensional aperture synthesis,” Astron. Astrophys. Suppl. 15, 427–430 (1974).

Bruck, Y. M.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Chalasinska-Macukov, K.

H. H. Arsenault, K. Chalasinska-Macukov, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983);K. Chalasinska-Macukov, H. Arsenault, “Fast iterative solution to exact equations for the two-dimensional phase problem,” J. Opt. Soc. Am. A 2, 46–50 (1985).
[CrossRef]

Coombs, J. A. G.

A. H. Greenaway, J. G. Walker, J. A. G. Coombs, “Ambiguities in speckle reconstructions. Some ways of avoiding them,” in Indirect Imaging, J. A. Roberts, ed. (Cambridge U. Press, Cambridge, 1984), pp. 111–117.

Currie, D. G.

B. R. Frieden, D. G. Currie, “On unfolding the autocorrelation function,” J. Opt. Soc. Am. 66, 1111 (A) (1976).

Dainty, J. C.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

Deighton, H. V.

Fiddy, M. A.

Fienup, J. R.

J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

J. R. Fienup, “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
[CrossRef]

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[CrossRef] [PubMed]

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

Frieden, B. R.

B. R. Frieden, D. G. Currie, “On unfolding the autocorrelation function,” J. Opt. Soc. Am. 66, 1111 (A) (1976).

Fuentes, F. J.

F. J. Fuentes, Instituto de Astrofísica de Canarias, La Laguna Tenerife, Spain (personal communication).

F. J. Fuentes, M. Nieto-Vesperinas, R. Navarro, “A fortan program to obtain a function of two variables from its autocorrelation,” submitted to Trans. Math. Software.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Greenaway, A. H.

A. H. Greenaway, J. G. Walker, J. A. G. Coombs, “Ambiguities in speckle reconstructions. Some ways of avoiding them,” in Indirect Imaging, J. A. Roberts, ed. (Cambridge U. Press, Cambridge, 1984), pp. 111–117.

Hammersley, J. M.

J. M. Hammersley, D. C. Handscomb, Monte Carlo Methods (Methuen, London, 1967), p. 113.

Handscomb, D. C.

J. M. Hammersley, D. C. Handscomb, Monte Carlo Methods (Methuen, London, 1967), p. 113.

Hayes, M. H.

P. L. Van Hove, M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1286–1293 (1983).
[CrossRef]

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Levi, A.

Lim, J. S.

P. L. Van Hove, M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1286–1293 (1983).
[CrossRef]

Liu, C. Y. C.

C. Y. C. Liu, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

Lohmann, A. W.

C. Y. C. Liu, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

McClellan, J. H.

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

Méndez, J. A.

M. Nieto-Vesperinas, J. A. Méndez, “Phase retrieval by Monte Carlo methods,” Opt. Commun. 59, 249–254 (1986).
[CrossRef]

Metropolis, N.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Napier, P. J.

P. J. Napier, R. H. T. Bates, “Inferring phase information from modulus information in two-dimensional aperture synthesis,” Astron. Astrophys. Suppl. 15, 427–430 (1974).

Navarro, R.

F. J. Fuentes, M. Nieto-Vesperinas, R. Navarro, “A fortan program to obtain a function of two variables from its autocorrelation,” submitted to Trans. Math. Software.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, J. A. Méndez, “Phase retrieval by Monte Carlo methods,” Opt. Commun. 59, 249–254 (1986).
[CrossRef]

M. Nieto-Vesperinas, “A study on the performance of least square optimization methods in the problem of phase retrieval,” Opt. Acta 33, 713–722 (1986).
[CrossRef]

F. J. Fuentes, M. Nieto-Vesperinas, R. Navarro, “A fortan program to obtain a function of two variables from its autocorrelation,” submitted to Trans. Math. Software.

Oppenheim, A. V.

P. L. Van Hove, M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1286–1293 (1983).
[CrossRef]

Paxman, R. G.

H. H. Barrett, W. E. Smith, R. G. Paxman, “Monte Carlo methods in optics,” Acta Polytech. Scand. Appl. Phys. Ser. 149, 3 (1985).

W. E. Smith, H. H. Barrett, R. G. Paxman, “Reconstruction of objects from coded images by simulated annealing,” Opt. Lett. 8, 199–201 (1983);W. E. Smith, R. G. Paxman, H. H. Barrett, “Image reconstruction from coded data: I. Reconstruction algorithms and experimental results,” J. Opt Soc. Am. A 2, 491–500 (1985).
[CrossRef] [PubMed]

Pincus, M.

M. Pincus, “A closed form solution of certain programming problems,” Oper. Res. 16, 690–694 (1968);“A Monte Carlo method for the approximate solution of certain types of constrained optimization problems,” Oper. Res. 18, 1225–1228 (1970).
[CrossRef]

Rosenbluth, A.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Rosenbluth, M.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Rubinstein, R. Y.

R. Y. Rubinstein, Simulation and the Monte Carlo Methods (Wiley, New York, 1981), p. 260.

Scivier, M. S.

Smith, W. E.

H. H. Barrett, W. E. Smith, R. G. Paxman, “Monte Carlo methods in optics,” Acta Polytech. Scand. Appl. Phys. Ser. 149, 3 (1985).

W. E. Smith, H. H. Barrett, R. G. Paxman, “Reconstruction of objects from coded images by simulated annealing,” Opt. Lett. 8, 199–201 (1983);W. E. Smith, R. G. Paxman, H. H. Barrett, “Image reconstruction from coded data: I. Reconstruction algorithms and experimental results,” J. Opt Soc. Am. A 2, 491–500 (1985).
[CrossRef] [PubMed]

Sodin, L. G.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Stark, H.

Teller, A.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Teller, E.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Van Hove, P. L.

P. L. Van Hove, M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1286–1293 (1983).
[CrossRef]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Wackerman, C. C.

Walker, J. G.

A. H. Greenaway, J. G. Walker, J. A. G. Coombs, “Ambiguities in speckle reconstructions. Some ways of avoiding them,” in Indirect Imaging, J. A. Roberts, ed. (Cambridge U. Press, Cambridge, 1984), pp. 111–117.

Acta Polytech. Scand. Appl. Phys. Ser.

H. H. Barrett, W. E. Smith, R. G. Paxman, “Monte Carlo methods in optics,” Acta Polytech. Scand. Appl. Phys. Ser. 149, 3 (1985).

Appl. Opt.

Astron. Astrophys. Suppl.

P. J. Napier, R. H. T. Bates, “Inferring phase information from modulus information in two-dimensional aperture synthesis,” Astron. Astrophys. Suppl. 15, 427–430 (1974).

IEEE Trans. Acoust. Speech Signal Process.

P. L. Van Hove, M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1286–1293 (1983).
[CrossRef]

J. Chem. Phys.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

J. Opt. Soc. Am.

B. R. Frieden, D. G. Currie, “On unfolding the autocorrelation function,” J. Opt. Soc. Am. 66, 1111 (A) (1976).

J. Opt. Soc. Am. A

Oper. Res.

M. Pincus, “A closed form solution of certain programming problems,” Oper. Res. 16, 690–694 (1968);“A Monte Carlo method for the approximate solution of certain types of constrained optimization problems,” Oper. Res. 18, 1225–1228 (1970).
[CrossRef]

Opt. Commun.

H. H. Arsenault, K. Chalasinska-Macukov, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983);K. Chalasinska-Macukov, H. Arsenault, “Fast iterative solution to exact equations for the two-dimensional phase problem,” J. Opt. Soc. Am. A 2, 46–50 (1985).
[CrossRef]

C. Y. C. Liu, A. W. Lohmann, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Opt. Lett.

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[CrossRef] [PubMed]

W. E. Smith, H. H. Barrett, R. G. Paxman, “Reconstruction of objects from coded images by simulated annealing,” Opt. Lett. 8, 199–201 (1983);W. E. Smith, R. G. Paxman, H. H. Barrett, “Image reconstruction from coded data: I. Reconstruction algorithms and experimental results,” J. Opt Soc. Am. A 2, 491–500 (1985).
[CrossRef] [PubMed]

Opt. Acta

M. Nieto-Vesperinas, “A study on the performance of least square optimization methods in the problem of phase retrieval,” Opt. Acta 33, 713–722 (1986).
[CrossRef]

Opt. Commun.

M. Nieto-Vesperinas, J. A. Méndez, “Phase retrieval by Monte Carlo methods,” Opt. Commun. 59, 249–254 (1986).
[CrossRef]

Opt. Eng.

J. R. Fienup, “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
[CrossRef]

Opt. Lett.

Optik

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik 61, 247–262 (1982);K. L. Garden, R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. II: One-dimensional considerations,” Optik 62, 131–142 (1982);W. R. Fright, R. M. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. III: Computational examples for two dimensions,” Optik 62, 219–230 (1982).

Proc. IEEE

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

Science

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Other

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

J. M. Hammersley, D. C. Handscomb, Monte Carlo Methods (Methuen, London, 1967), p. 113.

R. Y. Rubinstein, Simulation and the Monte Carlo Methods (Wiley, New York, 1981), p. 260.

F. J. Fuentes, M. Nieto-Vesperinas, R. Navarro, “A fortan program to obtain a function of two variables from its autocorrelation,” submitted to Trans. Math. Software.

A. H. Greenaway, J. G. Walker, J. A. G. Coombs, “Ambiguities in speckle reconstructions. Some ways of avoiding them,” in Indirect Imaging, J. A. Roberts, ed. (Cambridge U. Press, Cambridge, 1984), pp. 111–117.

F. J. Fuentes, Instituto de Astrofísica de Canarias, La Laguna Tenerife, Spain (personal communication).

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Figures (9)

Fig. 1
Fig. 1

Block diagram of the SA algorithm.

Fig. 2
Fig. 2

(a) Original object, which represents a 32 × 32 pixel low-pass-filtered version of object 5 of Ref. 21. (b) FA reconstruction from noiseless data.

Fig. 3
Fig. 3

(a) Autocorrelation and (b) power spectrum of the object shown in Plate 1(a).

Fig. 4
Fig. 4

(a) Autocorrelation and (b) power spectrum of the object shown in Plate 2(a).

Plate 1
Plate 1

Scale associating colors with object intensities.

Plate 2
Plate 2

(a) Original galaxy object compressed to 32 × 32 pixels. (b) FA reconstruction from noisy data with a S/N ratio of 10. (c) Low-pass-filtered version of (b). (d) SA reconstruction with a S/N ratio of 10. (e) Low-pass-filtered version of (d). (f) FA reconstruction with a S/N ratio of 4. (g) SA reconstruction with a S/N ratio of 4.

Plate 3
Plate 3

(a) FA reconstruction from simulated speckle interferometry data. (b) Low-pass-filtered version of (a). (c) SA reconstruction from simulated speckle interferometry data. (d) Low-pass-filtered version of (c). (e) Reconstruction combining FA and the SA algorithm from simulated speckle interferometry data. (f) Low-pass-filtered version of (e).

Plate 4
Plate 4

(a) Original two-nucleus quasar object compressed to 32 × 32 pixels. (b) FA reconstruction from noiseless data. (c) Reconstruction from noiseless data combining FA and the SA algorithm.

Plate 5
Plate 5

(a) FA reconstruction of the object shown in Plate 2(a) from noiseless data. (b) Reconstruction from noiseless data by combining FA and 10 repetitions of the SA algorithm.

Equations (10)

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Q i j = m = 1 M n = 1 M g m n g m + i M , n + j M , 1 m M , 1 m + i M M , 1 n + j M M , 1 n M ,
F ( g m n ) = [ k = 1 M 2 + ( M 1 ) 2 r j k 2 ] 1 / 2 , k = M ( i 1 ) + j ,
r k = Q i j m = 1 M n = 1 M g m n g m + i M , n + j M , 1 m + i M M , 1 n + j M M , k = M ( i 1 ) + j .
m = 1 M m = 1 M g m n g m + i M , n + j M
m = 1 M m = 1 M g m n g m + i M , n + j M
x * = l i m T 0 s d xx exp [ F ( x ) / T ] s d x exp [ F ( x ) / T ]
Δ F = F ( 1 ) F ( 0 ) .
Q ( m + 1 ) ( x , y ) = d ξ d η [ g ( m ) ( ξ , η ) + h δ ( ξ a , η b ) ] × [ g ( m ) ( η + x , η + y ) + h δ ( ξ + x a , η + y b ) ] = Q ( m ) ( x , y ) + h g ( m ) ( a x , b y ) + h g ( m ) ( a + x , b + y ) + h 2 δ ( x , y ) .
α n = B ( A + log F n C ) p
S / N ratio = | G ( u , υ ) | d u d υ / A σ ,

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