Abstract

An iterative phase retrieval method for nonperiodic objects has been developed from the charge-flipping algorithm proposed in crystallography. A combination of the hybrid input–output (HIO) algorithm and the flipping algorithm has greatly improved performance. In this combined algorithm the flipping algorithm serves to find the support (object boundary) dynamically, and the HIO part improves convergence and moves the algorithm out of local minima. It starts with a single intensity measurement in the Fourier domain and does not require a priori knowledge of the support in the image domain. This method is suitable for general image recovery from oversampled diffuse elastic x-ray and electron-diffraction intensities. The relationship between this algorithm and the output–output algorithm is elucidated.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. W. Gerchberg and W. O. Saxton, Optik (Stuttgart) 35, 237 (1972).
  2. J. R. Fienup, Appl. Opt. 21, 2758 (1982).
    [CrossRef] [PubMed]
  3. J. R. Fienup, Opt. Eng. 19, 297 (1980).
    [CrossRef]
  4. G. Oszlányi and A. Sütő, Acta Crystallogr. Sect. A 60, 134 (2004).
    [CrossRef]
  5. J. S. Wu, J. C. H. Spence, M. O’Keefe, and T. Groy, Acta Crystallogr. Sect. A,  60, 326 (2004).
    [CrossRef]
  6. R. P. Millane, J. Opt. Soc. Am. A 7, 394 (1990).
    [CrossRef]
  7. V. Elser, J. Opt. Soc. Am. A 20, 40 (2003).
    [CrossRef]
  8. J. R. Fienup, T. R. Crimmins, and W. Holsztynski, J. Opt. Soc. Am. 72, 610 (1982).
  9. T. R. Crimmins, J. R. Fienup, and B. J. Thelen, J. Opt. Soc. Am. A 7, 3 (1990).
    [CrossRef]
  10. S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. C. H. Spence, Phys. Rev. B 68, 140101 (2003).
    [CrossRef]

2004 (2)

G. Oszlányi and A. Sütő, Acta Crystallogr. Sect. A 60, 134 (2004).
[CrossRef]

J. S. Wu, J. C. H. Spence, M. O’Keefe, and T. Groy, Acta Crystallogr. Sect. A,  60, 326 (2004).
[CrossRef]

2003 (2)

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. C. H. Spence, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

V. Elser, J. Opt. Soc. Am. A 20, 40 (2003).
[CrossRef]

1990 (2)

1982 (2)

1980 (1)

J. R. Fienup, Opt. Eng. 19, 297 (1980).
[CrossRef]

1972 (1)

R. W. Gerchberg and W. O. Saxton, Optik (Stuttgart) 35, 237 (1972).

Chapman, H.

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. C. H. Spence, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Crimmins, T. R.

Elser, V.

Fienup, J. R.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, Optik (Stuttgart) 35, 237 (1972).

Groy, T.

J. S. Wu, J. C. H. Spence, M. O’Keefe, and T. Groy, Acta Crystallogr. Sect. A,  60, 326 (2004).
[CrossRef]

Hau-Riege, S.

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. C. H. Spence, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

He, H.

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. C. H. Spence, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Holsztynski, W.

Howells, M.

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. C. H. Spence, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Marchesini, S.

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. C. H. Spence, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Millane, R. P.

Noy, A.

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. C. H. Spence, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

O’Keefe, M.

J. S. Wu, J. C. H. Spence, M. O’Keefe, and T. Groy, Acta Crystallogr. Sect. A,  60, 326 (2004).
[CrossRef]

Oszlányi, G.

G. Oszlányi and A. Sütő, Acta Crystallogr. Sect. A 60, 134 (2004).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, Optik (Stuttgart) 35, 237 (1972).

Spence, J. C. H.

J. S. Wu, J. C. H. Spence, M. O’Keefe, and T. Groy, Acta Crystallogr. Sect. A,  60, 326 (2004).
[CrossRef]

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. C. H. Spence, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Süto, A.

G. Oszlányi and A. Sütő, Acta Crystallogr. Sect. A 60, 134 (2004).
[CrossRef]

Thelen, B. J.

Weierstall, U.

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. C. H. Spence, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Wu, J. S.

J. S. Wu, J. C. H. Spence, M. O’Keefe, and T. Groy, Acta Crystallogr. Sect. A,  60, 326 (2004).
[CrossRef]

Acta Crystallogr. Sect. A (2)

G. Oszlányi and A. Sütő, Acta Crystallogr. Sect. A 60, 134 (2004).
[CrossRef]

J. S. Wu, J. C. H. Spence, M. O’Keefe, and T. Groy, Acta Crystallogr. Sect. A,  60, 326 (2004).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Eng. (1)

J. R. Fienup, Opt. Eng. 19, 297 (1980).
[CrossRef]

Optik (Stuttgart) (1)

R. W. Gerchberg and W. O. Saxton, Optik (Stuttgart) 35, 237 (1972).

Phys. Rev. B (1)

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. C. H. Spence, Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

(a) Simulated modulus used for phase recovery test. (b) Image reconstructed by the charge-flipping algorithm alone. The reconstruction does not converge due to stagnation in a local minimum. (c) Reconstructed image with a combination of the flipping and the HIO algorithms. It reproduces the original image.

Fig. 2
Fig. 2

(a) Residual R factors as a function of iteration number for the reconstruction of Fig. 1(a) with the flipping algorithm. (b) R versus iteration number for the reconstruction in Fig. 1(c) with a combination of the flipping and the HIO algorithms. Although the HIO iterations show an increase in R due to an incorrectly reconstructed support, the stagnation problem that occurs with use of the flipping algorithm is solved.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

Πmodg=F-1FgFmodFgn,
Fout-out:gnmgnm+1=Πmodgmnif nS1-βΠmodgmnif nS,
Fflip:gnmgnm+1=Πmodgmnif nS-Πmodgmnif nS.
Fhybrid:gnmgnm+1=Πmodgmnif nSgnm-βΠmodgmnif nS.

Metrics