OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 15, Iss. 11 — Nov. 1, 1998
  • pp: 2849–2861

Study on the convergence property of the hybrid input–output algorithm used for phase retrieval

Hiroaki Takajo, Tohru Takahashi, Ryuzo Ueda, and Makoto Taninaka  »View Author Affiliations

JOSA A, Vol. 15, Issue 11, pp. 2849-2861 (1998)

View Full Text Article

Enhanced HTML    Acrobat PDF (357 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



Even though the hybrid input–output algorithm (HIO) has been recognized empirically to be one of the most successful versions of the iterative Fourier transform algorithm used for phase retrieval, its behavior is not yet well understood. Therefore a theoretical investigation on the convergence property of the HIO with an infinitesimally small feedback parameter is presented, although on a rather intuitive level, and it is shown that, until a solution is found, this algorithm continues to travel among the objects seeking those that satisfy the Fourier-domain constraint and for which the object-domain error has a locally minimum value. The concept of the territory is introduced with use of the algorithm constructed by modifying the HIO, and then the results are presented of the computer simulations for 2×2 objects with L-shaped support that were carried out to test the validity of our theory and to gain insight into the case in which the value of the feedback parameter is finite.

© 1998 Optical Society of America

OCIS Codes
(070.2590) Fourier optics and signal processing : ABCD transforms
(100.2000) Image processing : Digital image processing
(100.3010) Image processing : Image reconstruction techniques
(100.5070) Image processing : Phase retrieval

Original Manuscript: February 13, 1998
Revised Manuscript: July 23, 1998
Manuscript Accepted: August 10, 1998
Published: November 1, 1998

Hiroaki Takajo, Tohru Takahashi, Ryuzo Ueda, and Makoto Taninaka, "Study on the convergence property of the hybrid input–output algorithm used for phase retrieval," J. Opt. Soc. Am. A 15, 2849-2861 (1998)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, 1987).
  2. H. H. Arsenault, K. Chalasinska-Macukow, “The solution of the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983). [CrossRef]
  3. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  4. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
  5. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986). [CrossRef]
  6. J. H. Seldin, J. R. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7, 412–427 (1990). [CrossRef]
  7. C. C. Wackerman, A. E. Yagle, “Use of Fourier domain real-plane zeros to overcome a phase retrieval stagnation,” J. Opt. Soc. Am. A 8, 1898–1904 (1991). [CrossRef]
  8. C. C. Wackerman, A. E. Yagle, “Phase retrieval and estimation with use of real-plane zeros,” J. Opt. Soc. Am. A 11, 2016–2026 (1994). [CrossRef]
  9. P.-T. Chen, M. A. Fiddy, “Image reconstruction from power spectral data with use of point-zero locations,” J. Opt. Soc. Am. A 11, 2210–2214 (1994). [CrossRef]
  10. P.-T. Chen, M. A. Fiddy, C.-W. Liao, D. A. Pommet, “Blind deconvolution and phase retrieval from point zeros,” J. Opt. Soc. Am. A 13, 1524–1531 (1996). [CrossRef]
  11. R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–526 (1987). [CrossRef]
  12. C. R. Parker, P. J. Bones, “Convergence of iterative phase retrieval improved by utilizing zero sheets,” Opt. Commun. 92, 209–214 (1992). [CrossRef]
  13. P. J. Bones, C. R. Parker, B. L. Satherley, R. W. Watson, “Deconvolution and phase retrieval with use of zero sheets,” J. Opt. Soc. Am. A 12, 1842–1857 (1995). [CrossRef]
  14. H. Takajo, T. Takahashi, H. Kawanami, R. Ueda, “Numerical investigation of the iterative phase-retrieval stagnation problem: territories of convergence objects and holes in their boundaries,” J. Opt. Soc. Am. A 14, 3175–3187 (1997). [CrossRef]
  15. R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987). [CrossRef]
  16. H. Takajo, T. Takahashi, “Competence of hybrid input–output algorithm for phase retrieval,” Jpn. J. Opt. 22, 419–427 (1993).

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.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited