Abstract

Based on the minimization of the Lagrange formula, which is composed of two kinds of information measure, the maximum entropy method (MEM) is derived for diffractive imaging contaminated by quantum noise. This gives a suitable object corresponding to the maximum entropy principle with an iterative procedure. The MEM-based iterative phase retrieval algorithm with the initial process of the hybrid input–output (HIO-MEM) is presented, and a simple numerical example shows that the algorithm is effective for Poisson noise added to Fourier intensity. The relationship between the newly derived MEM for diffractive imaging and the conventional MEM for structure analysis based on crystallography is revealed.

© 2008 Optical Society of America

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  1. D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5, 843 (1952).
    [CrossRef]
  2. 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).
  3. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758-2769 (1982).
    [CrossRef] [PubMed]
  4. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature (London) 400, 342-344 (1999).
    [CrossRef]
  5. J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
    [CrossRef] [PubMed]
  6. J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (2003).
    [CrossRef]
  7. Y. Nishino, J. Miao, and T. Ishikawa, “Image reconstruction of nanostructured nonperiodic objects only from oversampled hard x-ray diffraction intensities,” Phys. Rev. B 68, 220101(R) (2003); http://www.nature.com/nphys/journal/v2/n12/index.html (October 2008).
    [CrossRef]
  8. H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839-843 (2006).
    [CrossRef]
  9. U. Weierstall, Q. Chen, J. C. H. Spence, M. R. Howells, M. Isaacson, and R. R. Panepucci, “Image reconstruction from electron and X-ray diffraction patterns using iterative algorithms: experiment and simulation,” Ultramicroscopy 90, 171-195 (2002).
    [CrossRef] [PubMed]
  10. J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419-1421 (2003).
    [CrossRef] [PubMed]
  11. O. Kamimura, K. Kawahara, T. Doi, T. Dobashi, T. Abe, and K. Gohara, “Diffraction microscopy using 20 kV electron beam for multiwall carbon nanotubes,” Appl. Phys. Lett. 92, 024106 (2008).
    [CrossRef]
  12. R. L. Sandberg, “Lensless diffractive imaging using tabletop coherent high-harmonic soft-x-ray beams,” Phys. Rev. Lett. 99, 098103 (2007).
    [CrossRef] [PubMed]
  13. J. C. H. Spence, “Diffractive (Lensless) imaging,” in Science of Microscopy, P.W.Hawkes and J.C. H.Spence, eds. (Springer, 2007), Chap. 19.
  14. R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394-411 (1990).
    [CrossRef]
  15. D. M. Collins, “Electron density images from imperfect data by iterative entropy maximization,” Nature (London) 298, 49-51 (1982).
    [CrossRef]
  16. S. F. Gull and G. J. Daniell, “Image reconstruction from incomplete and noisy data,” Nature (London) 272, 686-690 (1978).
    [CrossRef]
  17. S. F. Gull, A. K. Livesey, and D. S. Sivia, “Maximum entropy solution of a small centrosymmetric crystal structure,” Acta Crystallogr. 43, 112-117 (1987).
    [CrossRef]
  18. B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 511 (1972).
    [CrossRef] [PubMed]
  19. O. E. Piro, “Information theory and the 'phase problem' in crystallography,” Acta Crystallogr. A39, 61-68 (1983).
  20. W. Wei, “Application of the maximum entropy method to electron density determination,” J. Appl. Crystallogr. 18, 442-445 (1985).
    [CrossRef]
  21. A. Podjarny, D. Moras, J. Navaza, and P. M. Alizari, “Low-resolution phase extension and refinement by maximum entropy,” Acta Crystallogr. A44, 545-551 (1988).
  22. M. Sakata and M. Sato, “Accurate structure analysis by the maximum-entropy method,” Acta Crystallogr. 46, 63 (1990).
  23. H. Shioya and K. Gohara, “Generalized phase retrieval algorithm based on information measures,” Opt. Commun. 266, 88-93 (2006).
    [CrossRef]
  24. K. Choi and A. Lanterman, “Phase retrieval from noisy data based on minimization of penalized I-divergence,” J. Opt. Soc. Am. A 24, 34-49 (2007).
    [CrossRef]
  25. C. E. Shannon, 'A mathematical theory of communication,' Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).
  26. S. Kullback and R. A. Leibler, “On information and sufficiency,”Ann. Math. Stat. 22, 79-86 (1951).
    [CrossRef]
  27. I. Csiszár, “On topological properties of f-divergences,” Stud. Sci. Math. Hung. 2, 329-339 (1967).
  28. I. Csiszár, “Why least squares and maximum entropy?An axiomatic approach to inverse problems,” Ann. Stat. 19, 2033-2066 (1991).
    [CrossRef]
  29. G. Oszlanyi and A. Suto, “Ab initio structure solution by charge flipping,” Acta Crystallogr. 60, 134-141 (2004).
    [CrossRef]
  30. E. T. Jayes, “Information theory and statistical mechanics,” Phys. Rev. 106, 620-630 (1957).
    [CrossRef]
  31. E. T. Jayes, “Information theory and statistical mechanics. II,” Phys. Rev. 108, 171-175 (1957).
    [CrossRef]

2008 (1)

O. Kamimura, K. Kawahara, T. Doi, T. Dobashi, T. Abe, and K. Gohara, “Diffraction microscopy using 20 kV electron beam for multiwall carbon nanotubes,” Appl. Phys. Lett. 92, 024106 (2008).
[CrossRef]

2007 (2)

R. L. Sandberg, “Lensless diffractive imaging using tabletop coherent high-harmonic soft-x-ray beams,” Phys. Rev. Lett. 99, 098103 (2007).
[CrossRef] [PubMed]

K. Choi and A. Lanterman, “Phase retrieval from noisy data based on minimization of penalized I-divergence,” J. Opt. Soc. Am. A 24, 34-49 (2007).
[CrossRef]

2006 (2)

H. Shioya and K. Gohara, “Generalized phase retrieval algorithm based on information measures,” Opt. Commun. 266, 88-93 (2006).
[CrossRef]

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839-843 (2006).
[CrossRef]

2004 (1)

G. Oszlanyi and A. Suto, “Ab initio structure solution by charge flipping,” Acta Crystallogr. 60, 134-141 (2004).
[CrossRef]

2003 (3)

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (2003).
[CrossRef]

Y. Nishino, J. Miao, and T. Ishikawa, “Image reconstruction of nanostructured nonperiodic objects only from oversampled hard x-ray diffraction intensities,” Phys. Rev. B 68, 220101(R) (2003); http://www.nature.com/nphys/journal/v2/n12/index.html (October 2008).
[CrossRef]

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

2002 (2)

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

U. Weierstall, Q. Chen, J. C. H. Spence, M. R. Howells, M. Isaacson, and R. R. Panepucci, “Image reconstruction from electron and X-ray diffraction patterns using iterative algorithms: experiment and simulation,” Ultramicroscopy 90, 171-195 (2002).
[CrossRef] [PubMed]

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature (London) 400, 342-344 (1999).
[CrossRef]

1991 (1)

I. Csiszár, “Why least squares and maximum entropy?An axiomatic approach to inverse problems,” Ann. Stat. 19, 2033-2066 (1991).
[CrossRef]

1990 (2)

M. Sakata and M. Sato, “Accurate structure analysis by the maximum-entropy method,” Acta Crystallogr. 46, 63 (1990).

R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394-411 (1990).
[CrossRef]

1988 (1)

A. Podjarny, D. Moras, J. Navaza, and P. M. Alizari, “Low-resolution phase extension and refinement by maximum entropy,” Acta Crystallogr. A44, 545-551 (1988).

1987 (1)

S. F. Gull, A. K. Livesey, and D. S. Sivia, “Maximum entropy solution of a small centrosymmetric crystal structure,” Acta Crystallogr. 43, 112-117 (1987).
[CrossRef]

1985 (1)

W. Wei, “Application of the maximum entropy method to electron density determination,” J. Appl. Crystallogr. 18, 442-445 (1985).
[CrossRef]

1983 (1)

O. E. Piro, “Information theory and the 'phase problem' in crystallography,” Acta Crystallogr. A39, 61-68 (1983).

1982 (2)

D. M. Collins, “Electron density images from imperfect data by iterative entropy maximization,” Nature (London) 298, 49-51 (1982).
[CrossRef]

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

1978 (1)

S. F. Gull and G. J. Daniell, “Image reconstruction from incomplete and noisy data,” Nature (London) 272, 686-690 (1978).
[CrossRef]

1972 (2)

B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 511 (1972).
[CrossRef] [PubMed]

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).

1967 (1)

I. Csiszár, “On topological properties of f-divergences,” Stud. Sci. Math. Hung. 2, 329-339 (1967).

1957 (2)

E. T. Jayes, “Information theory and statistical mechanics,” Phys. Rev. 106, 620-630 (1957).
[CrossRef]

E. T. Jayes, “Information theory and statistical mechanics. II,” Phys. Rev. 108, 171-175 (1957).
[CrossRef]

1952 (1)

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5, 843 (1952).
[CrossRef]

1951 (1)

S. Kullback and R. A. Leibler, “On information and sufficiency,”Ann. Math. Stat. 22, 79-86 (1951).
[CrossRef]

1948 (1)

C. E. Shannon, 'A mathematical theory of communication,' Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).

Abe, T.

O. Kamimura, K. Kawahara, T. Doi, T. Dobashi, T. Abe, and K. Gohara, “Diffraction microscopy using 20 kV electron beam for multiwall carbon nanotubes,” Appl. Phys. Lett. 92, 024106 (2008).
[CrossRef]

Alizari, P. M.

A. Podjarny, D. Moras, J. Navaza, and P. M. Alizari, “Low-resolution phase extension and refinement by maximum entropy,” Acta Crystallogr. A44, 545-551 (1988).

Anderson, E. H.

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (2003).
[CrossRef]

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Barty, A.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839-843 (2006).
[CrossRef]

Bogan, M. J.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839-843 (2006).
[CrossRef]

Boutet, S.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839-843 (2006).
[CrossRef]

Chapman, H. N.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839-843 (2006).
[CrossRef]

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature (London) 400, 342-344 (1999).
[CrossRef]

Chen, Q.

U. Weierstall, Q. Chen, J. C. H. Spence, M. R. Howells, M. Isaacson, and R. R. Panepucci, “Image reconstruction from electron and X-ray diffraction patterns using iterative algorithms: experiment and simulation,” Ultramicroscopy 90, 171-195 (2002).
[CrossRef] [PubMed]

Choi, K.

Collins, D. M.

D. M. Collins, “Electron density images from imperfect data by iterative entropy maximization,” Nature (London) 298, 49-51 (1982).
[CrossRef]

Csiszár, I.

I. Csiszár, “Why least squares and maximum entropy?An axiomatic approach to inverse problems,” Ann. Stat. 19, 2033-2066 (1991).
[CrossRef]

I. Csiszár, “On topological properties of f-divergences,” Stud. Sci. Math. Hung. 2, 329-339 (1967).

Daniell, G. J.

S. F. Gull and G. J. Daniell, “Image reconstruction from incomplete and noisy data,” Nature (London) 272, 686-690 (1978).
[CrossRef]

Dobashi, T.

O. Kamimura, K. Kawahara, T. Doi, T. Dobashi, T. Abe, and K. Gohara, “Diffraction microscopy using 20 kV electron beam for multiwall carbon nanotubes,” Appl. Phys. Lett. 92, 024106 (2008).
[CrossRef]

Doi, T.

O. Kamimura, K. Kawahara, T. Doi, T. Dobashi, T. Abe, and K. Gohara, “Diffraction microscopy using 20 kV electron beam for multiwall carbon nanotubes,” Appl. Phys. Lett. 92, 024106 (2008).
[CrossRef]

Fienup, J. R.

Frank, M.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839-843 (2006).
[CrossRef]

Frieden, B. R.

Gao, M.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Gerchberg, R. W.

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).

Gohara, K.

O. Kamimura, K. Kawahara, T. Doi, T. Dobashi, T. Abe, and K. Gohara, “Diffraction microscopy using 20 kV electron beam for multiwall carbon nanotubes,” Appl. Phys. Lett. 92, 024106 (2008).
[CrossRef]

H. Shioya and K. Gohara, “Generalized phase retrieval algorithm based on information measures,” Opt. Commun. 266, 88-93 (2006).
[CrossRef]

Gull, S. F.

S. F. Gull, A. K. Livesey, and D. S. Sivia, “Maximum entropy solution of a small centrosymmetric crystal structure,” Acta Crystallogr. 43, 112-117 (1987).
[CrossRef]

S. F. Gull and G. J. Daniell, “Image reconstruction from incomplete and noisy data,” Nature (London) 272, 686-690 (1978).
[CrossRef]

Hau-Riege, S. P.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839-843 (2006).
[CrossRef]

Hodgson, K. O.

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (2003).
[CrossRef]

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Howells, M. R.

U. Weierstall, Q. Chen, J. C. H. Spence, M. R. Howells, M. Isaacson, and R. R. Panepucci, “Image reconstruction from electron and X-ray diffraction patterns using iterative algorithms: experiment and simulation,” Ultramicroscopy 90, 171-195 (2002).
[CrossRef] [PubMed]

Isaacson, M.

U. Weierstall, Q. Chen, J. C. H. Spence, M. R. Howells, M. Isaacson, and R. R. Panepucci, “Image reconstruction from electron and X-ray diffraction patterns using iterative algorithms: experiment and simulation,” Ultramicroscopy 90, 171-195 (2002).
[CrossRef] [PubMed]

Ishikawa, T.

Y. Nishino, J. Miao, and T. Ishikawa, “Image reconstruction of nanostructured nonperiodic objects only from oversampled hard x-ray diffraction intensities,” Phys. Rev. B 68, 220101(R) (2003); http://www.nature.com/nphys/journal/v2/n12/index.html (October 2008).
[CrossRef]

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (2003).
[CrossRef]

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Jayes, E. T.

E. T. Jayes, “Information theory and statistical mechanics,” Phys. Rev. 106, 620-630 (1957).
[CrossRef]

E. T. Jayes, “Information theory and statistical mechanics. II,” Phys. Rev. 108, 171-175 (1957).
[CrossRef]

Johnson, B.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Kamimura, O.

O. Kamimura, K. Kawahara, T. Doi, T. Dobashi, T. Abe, and K. Gohara, “Diffraction microscopy using 20 kV electron beam for multiwall carbon nanotubes,” Appl. Phys. Lett. 92, 024106 (2008).
[CrossRef]

Kawahara, K.

O. Kamimura, K. Kawahara, T. Doi, T. Dobashi, T. Abe, and K. Gohara, “Diffraction microscopy using 20 kV electron beam for multiwall carbon nanotubes,” Appl. Phys. Lett. 92, 024106 (2008).
[CrossRef]

Kirz, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature (London) 400, 342-344 (1999).
[CrossRef]

Kullback, S.

S. Kullback and R. A. Leibler, “On information and sufficiency,”Ann. Math. Stat. 22, 79-86 (1951).
[CrossRef]

Lai, B.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Lanterman, A.

Leibler, R. A.

S. Kullback and R. A. Leibler, “On information and sufficiency,”Ann. Math. Stat. 22, 79-86 (1951).
[CrossRef]

Livesey, A. K.

S. F. Gull, A. K. Livesey, and D. S. Sivia, “Maximum entropy solution of a small centrosymmetric crystal structure,” Acta Crystallogr. 43, 112-117 (1987).
[CrossRef]

Miao, J.

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (2003).
[CrossRef]

Y. Nishino, J. Miao, and T. Ishikawa, “Image reconstruction of nanostructured nonperiodic objects only from oversampled hard x-ray diffraction intensities,” Phys. Rev. B 68, 220101(R) (2003); http://www.nature.com/nphys/journal/v2/n12/index.html (October 2008).
[CrossRef]

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature (London) 400, 342-344 (1999).
[CrossRef]

Millane, R. P.

Moras, D.

A. Podjarny, D. Moras, J. Navaza, and P. M. Alizari, “Low-resolution phase extension and refinement by maximum entropy,” Acta Crystallogr. A44, 545-551 (1988).

Nagahara, L. A.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Navaza, J.

A. Podjarny, D. Moras, J. Navaza, and P. M. Alizari, “Low-resolution phase extension and refinement by maximum entropy,” Acta Crystallogr. A44, 545-551 (1988).

Nishino, Y.

Y. Nishino, J. Miao, and T. Ishikawa, “Image reconstruction of nanostructured nonperiodic objects only from oversampled hard x-ray diffraction intensities,” Phys. Rev. B 68, 220101(R) (2003); http://www.nature.com/nphys/journal/v2/n12/index.html (October 2008).
[CrossRef]

Oszlanyi, G.

G. Oszlanyi and A. Suto, “Ab initio structure solution by charge flipping,” Acta Crystallogr. 60, 134-141 (2004).
[CrossRef]

Panepucci, R. R.

U. Weierstall, Q. Chen, J. C. H. Spence, M. R. Howells, M. Isaacson, and R. R. Panepucci, “Image reconstruction from electron and X-ray diffraction patterns using iterative algorithms: experiment and simulation,” Ultramicroscopy 90, 171-195 (2002).
[CrossRef] [PubMed]

Piro, O. E.

O. E. Piro, “Information theory and the 'phase problem' in crystallography,” Acta Crystallogr. A39, 61-68 (1983).

Podjarny, A.

A. Podjarny, D. Moras, J. Navaza, and P. M. Alizari, “Low-resolution phase extension and refinement by maximum entropy,” Acta Crystallogr. A44, 545-551 (1988).

Sakata, M.

M. Sakata and M. Sato, “Accurate structure analysis by the maximum-entropy method,” Acta Crystallogr. 46, 63 (1990).

Sandberg, R. L.

R. L. Sandberg, “Lensless diffractive imaging using tabletop coherent high-harmonic soft-x-ray beams,” Phys. Rev. Lett. 99, 098103 (2007).
[CrossRef] [PubMed]

Sato, M.

M. Sakata and M. Sato, “Accurate structure analysis by the maximum-entropy method,” Acta Crystallogr. 46, 63 (1990).

Saxton, W. O.

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).

Sayre, D.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature (London) 400, 342-344 (1999).
[CrossRef]

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5, 843 (1952).
[CrossRef]

Shannon, C. E.

C. E. Shannon, 'A mathematical theory of communication,' Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).

Shioya, H.

H. Shioya and K. Gohara, “Generalized phase retrieval algorithm based on information measures,” Opt. Commun. 266, 88-93 (2006).
[CrossRef]

Sivia, D. S.

S. F. Gull, A. K. Livesey, and D. S. Sivia, “Maximum entropy solution of a small centrosymmetric crystal structure,” Acta Crystallogr. 43, 112-117 (1987).
[CrossRef]

Spence, J. C. H.

U. Weierstall, Q. Chen, J. C. H. Spence, M. R. Howells, M. Isaacson, and R. R. Panepucci, “Image reconstruction from electron and X-ray diffraction patterns using iterative algorithms: experiment and simulation,” Ultramicroscopy 90, 171-195 (2002).
[CrossRef] [PubMed]

J. C. H. Spence, “Diffractive (Lensless) imaging,” in Science of Microscopy, P.W.Hawkes and J.C. H.Spence, eds. (Springer, 2007), Chap. 19.

Suto, A.

G. Oszlanyi and A. Suto, “Ab initio structure solution by charge flipping,” Acta Crystallogr. 60, 134-141 (2004).
[CrossRef]

Vartanyants, I.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Wei, W.

W. Wei, “Application of the maximum entropy method to electron density determination,” J. Appl. Crystallogr. 18, 442-445 (1985).
[CrossRef]

Weierstall, U.

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Figures (3)

Fig. 1
Fig. 1

Gerchberg–Saxton iterative diagram of phase retrieval based on two function spaces, F O and F K .

Fig. 2
Fig. 2

ρ er and ρ mem are obtained by the HIO-ER and HIO-MEM, respectively. The figure on the upper left represents ρ target .

Fig. 3
Fig. 3

Discrimination based on I-divergence between two object functions is shown in the function space F O . The I-divergence I mem (from ρ target to ρ mem ) is less than I er (from ρ target to ρ er ) with a suitable setting of the parameters of the HIO-MEM.

Equations (11)

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L = D + λ E ,
L = I ( ρ n + 1 , ρ n ) + λ E ( F n + 1 , F n + 1 ) ,
I ( ρ n + 1 , ρ n ) = r ρ n + 1 ( r ) ln ρ n + 1 ( r ) ρ n ( r ) + r ρ n ( r ) r ρ n + 1 ( r ) ,
E ( F n + 1 , F n + 1 ) = 1 M k F n + 1 ( k ) F n + 1 ( k ) 2 ,
E ( F n + 1 , F n + 1 ) ρ n + 1 = 2 M ( ρ n + 1 ρ n + 1 ) .
ρ n + 1 ( r ) = ρ n ( r ) exp { ξ ( ρ n ( r ) ρ n ( r ) ) } .
ρ n + 1 ( r ) = { ρ n ( r ) r S , n n 0 , ρ n ( r ) β ρ n ( r ) r S , ρ n ( r ) exp { ξ ( ρ n ( r ) ρ n ( r ) ) } r S , n > n 0 , 0 r S , }
ρ n + 1 ( r ) = { ρ n ( r ) r S , n n 0 , ρ n ( r ) β ρ n ( r ) r S , ρ n ( r ) r S , n > n 0 . 0 r S , }
Poisson { F target c ( k ) 2 } c 2 F obs poisson ( k ) 2 ,
Q ( λ ) = r ρ ( r ) ln ρ ( r ) τ ( r ) λ 2 k F cal ( k ) F obs ( k ) exp [ i ψ model ( k ) ] 2 σ ( k ) 2
ρ ( r ) = exp { ln τ ( r ) + λ F 0 k ( F obs ( k ) exp [ i ψ model ( k ) ] F cal ( k ) ) σ ( k ) 2 exp { i 2 π k r } } .

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