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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 16, Iss. 9 — Sep. 1, 1999
  • pp: 2163–2168

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

Hiroaki Takajo, Tohru Takahashi, and Takao Shizuma  »View Author Affiliations

JOSA A, Vol. 16, Issue 9, pp. 2163-2168 (1999)

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In our previous paper [J. Opt. Soc. Am. A 15, 2849 (1998)], we investigated the behavior of the hybrid input–output algorithm (HIO) used for phase retrieval and showed that the HIO with an infinitesimally small feedback parameter has two abilities that are extremely important in finding a solution: the ability to locate an output-stagnation object and the ability to emerge from an output-stagnation object if it is not a solution. The behavior of the HIO is analyzed further, and an additional aspect is demonstrated that is also important in finding a solution. That is, outside the support of the input object of the HIO, the output-object values outside the support that have been produced so far accumulate so that the probability is conjectured to be extremely low that the HIO will fall into a periodic state. On the other hand, it is also demonstrated that this aspect of the HIO is not usually effective in the combination algorithm composed of the HIO and the error-reduction algorithm. To give credibility to our arguments, we show results that were obtained by reexamining the examples adopted in Seldin and Fienup’s numerical experiments [J. Opt. Soc. Am. A 7, 412 (1990)].

© 1999 Optical Society of America

OCIS Codes
(100.2000) Image processing : Digital image processing
(100.3010) Image processing : Image reconstruction techniques
(100.5070) Image processing : Phase retrieval

Original Manuscript: November 19, 1998
Revised Manuscript: April 5, 1999
Manuscript Accepted: April 5, 1999
Published: September 1, 1999

Hiroaki Takajo, Tohru Takahashi, and Takao Shizuma, "Further study on the convergence property of the hybrid input–output algorithm used for phase retrieval," J. Opt. Soc. Am. A 16, 2163-2168 (1999)

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  1. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
  2. R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987). [CrossRef]
  3. C. R. Parker, P. J. Bones, “Convergence of iterative phase retrieval improved by utilizing zero sheets,” Opt. Commun. 92, 209–214 (1992). [CrossRef]
  4. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986). [CrossRef]
  5. J. H. Seldin, J. R. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7, 412–427 (1990). [CrossRef]
  6. 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]
  7. H. Takajo, T. Takahashi, R. Ueda, M. 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). [CrossRef]
  8. Recall that, in Ref. 7, we developed, by modifying the HIO, an algorithm that was called the converging part of the HIO and was abbreviated as CPHIO [see Eqs. (41) and (42) of Ref. 7]. As was demonstrated in that reference, the CPHIO can converge to one of the output-stagnation objects of the HIO even when the value of β is finite. Therefore the CPHIO can relate any object in the MN-dimensional space to some output-stagnation object in the sense that, if the CPHIO starts from the object, it reaches the output-stagnation object. The territory of an output-stagnation object was defined as the subspace in the MN-dimensional space that is formed by the set of initial input objects related to the output-stagnation object in this sense.

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