OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 15, Iss. 6 — Jun. 1, 1998
  • pp: 1662–1669

Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects

J. Miao, D. Sayre, and H. N. Chapman  »View Author Affiliations

JOSA A, Vol. 15, Issue 6, pp. 1662-1669 (1998)

View Full Text Article

Enhanced HTML    Acrobat PDF (1389 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



It is suggested that, given the magnitude of Fourier transforms sampled at the Bragg density, the phase problem is underdetermined by a factor of 2 for 1D, 2D, and 3D objects. It is therefore unnecessary to oversample the magnitude of Fourier transforms by 2× in each dimension (i.e., oversampling by 4× for 2D and 8× for 3D) in retrieving the phase of 2D and 3D objects. Our computer phasing experiments accurately retrieved the phase from the magnitude of the Fourier transforms of 2D and 3D complex-valued objects by using positivity constraints on the imaginary part of the objects and loose supports, with the oversampling factor much less than 4 for 2D and 8 for 3D objects. Under the same conditions we also obtained reasonably good reconstructions of 2D and 3D complex-valued objects from the magnitude of their Fourier transforms with added noise and a central stop.

© 1998 Optical Society of America

OCIS Codes
(070.2590) Fourier optics and signal processing : ABCD transforms
(100.0100) Image processing : Image processing
(100.5070) Image processing : Phase retrieval

Original Manuscript: October 2, 1997
Revised Manuscript: January 15, 1998
Manuscript Accepted: January 21, 1998
Published: June 1, 1998

J. Miao, D. Sayre, and H. N. Chapman, "Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects," J. Opt. Soc. Am. A 15, 1662-1669 (1998)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  2. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978). [CrossRef] [PubMed]
  3. J. R. Fienup, “Phase retrieval algorithm: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
  4. R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase reconstruction procedure,” J. Opt. Soc. Am. 73, 358–365 (1983). [CrossRef]
  5. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986). [CrossRef]
  6. R. H. T. Bates, M. J. McDonnell, Image restoration and reconstruction (Oxford U. Press, Oxford, UK, 1986).
  7. D. Sayre, H. N. Chapman, J. Miao, “On the possible extension of x-ray crystallography to non-crystals,” Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. (to be published).
  8. J. Miao, H. N. Chapman, D. Sayre, “Image reconstruction from the oversampled diffraction pattern,” Microscopy Microanalysis 3 (Suppl. 2), 1155–1156 (1997).
  9. S. Lindaas, M. Howells, C. Jacobsen, A. Kalinovsky, “X-ray holographic microscopy by means of photoresist recording and atomic-force microscope readout,” J. Opt. Soc. Am. A 13, 1788–1800 (1996). [CrossRef]
  10. Yu. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979). [CrossRef]
  11. M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-30, 140–154 (1982). [CrossRef]
  12. J. Boyes-Watson, K. Davidson, M. F. Perutz, “An x-ray study of horse methaemoglobin. I,” Proc. R. Soc. London, Ser. A 191, 83–137 (1947). [CrossRef]
  13. D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5, 843 (1952). [CrossRef]
  14. R. H. T. Bates, “Fourier phase problems are uniquely soluble in more than one dimension. I: underlying theory,” Optik (Stuttgart) 61, 247–262 (1982).
  15. R. H. T. Bates, “Uniqueness of solution to two-dimensional Fourier phase problems of localized and positive images,” Comput. Vis. Graph. Image Process. 25, 205–217 (1984). [CrossRef]
  16. R. P. Millane, W. J. Stroud, “Reconstructing symmetric images from their undersampled Fourier intensities,” J. Opt. Soc. Am. A 14, 568–579 (1997). [CrossRef]
  17. A. Szöke, University of California, Lawrence Livermore National Laboratory, P.O. Box 808, L-41, Livermore, Calif. 94551 (personal communication, December1995). Szöke said that he thought 2× oversampling should in theory suffice for any dimensionality ⩾2.
  18. R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase recovery,” J. Math. Phys. 25, 3190–3193 (1984). [CrossRef]
  19. R. H. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. Bates, eds., Proc. SPIE558, 54–59 (1985). [CrossRef]
  20. R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987). [CrossRef]
  21. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987). [CrossRef]
  22. D. Sayre, H. N. Chapman, “X-ray microscopy,” Acta Crystallogr. Sect. A 51, 237–252 (1995). [CrossRef]
  23. G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Wiley, New York, 1989).
  24. B. L. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50,-30,000 eV,Z=1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993). [CrossRef]
  25. J. Kirz, C. Jacobsen, M. Howells, “Soft x-ray microscopes and their biological applications,” Q. Rev. Biophys. 28, 33–130 (1995). [CrossRef] [PubMed]
  26. The exception is where the material amplifies the incident x-ray beam, as with the x-ray laser amplifier.

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
Fig. 4 Fig. 5

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited