OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Stephen A. Burns
  • Vol. 25, Iss. 11 — Nov. 1, 2008
  • pp: 2846–2850

Maximum entropy method for diffractive imaging

Hiroyuki Shioya and Kazutoshi Gohara  »View Author Affiliations

JOSA A, Vol. 25, Issue 11, pp. 2846-2850 (2008)

View Full Text Article

Enhanced HTML    Acrobat PDF (236 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

OCIS Codes
(100.5070) Image processing : Phase retrieval
(100.3008) Image processing : Image recognition, algorithms and filters

ToC Category:
Image Processing

Original Manuscript: March 18, 2008
Revised Manuscript: September 8, 2008
Manuscript Accepted: September 8, 2008
Published: October 24, 2008

Hiroyuki Shioya and Kazutoshi Gohara, "Maximum entropy method for diffractive imaging," J. Opt. Soc. Am. A 25, 2846-2850 (2008)

Sort:  Author  |  Year  |  Journal  |  Reset  


  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]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


Fig. 1 Fig. 2 Fig. 3

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited