## Reconstruction of an Object from its Fourier Modulus: Development of the Combination Algorithm Composed of the Hybrid Input–Output Algorithm and Its Converging Part

Applied Optics, Vol. 41, Issue 29, pp. 6143-6153 (2002)

http://dx.doi.org/10.1364/AO.41.006143

Acrobat PDF (203 KB)

### Abstract

The hybrid input-output algorithm (HIO) used for phase retrieval is in many cases combined with the error-reduction algorithm (ER) to attempt to stabilize the HIO. However, in our previous paper [J. Opt. Soc. Am. A <b>16,</b> 2163 (1999)], it was demonstrated that this combination makes it more likely that the resultant algorithm will fall into a periodic state before reaching a solution because the values of the input object outside the support, which is imposed as the object-domain constraint, are set to be zero in the intervals in which the ER is implemented. This paper deals with this problem inherent in the combination algorithm. The converging part of the HIO (CPHIO), which is an algorithm we previously developed [J. Opt. Soc. Am. A <b>15,</b> 2849 (1998)], can be thought of as an extension of the ER for the case in which the input object can have nonzero values outside the support. Keeping this in mind, the algorithm is then constructed by combining the HIO with the CPHIO instead of with the ER. The computer simulation results that demonstrate the effectiveness of the proposed algorithm are given.

© 2002 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

**Citation**

Hiroaki Takajo, Tohru Takahashi, Katsuhiko Itoh, and Toshiro Fujisaki, "Reconstruction of an Object from its Fourier Modulus: Development of the Combination Algorithm Composed of the Hybrid Input–Output Algorithm and Its Converging Part," Appl. Opt. **41**, 6143-6153 (2002)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-41-29-6143

Sort: Year | Journal | Reset

### References

- J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
- J. C. Dainty and J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, Calif., 1987), Chap. 7, pp. 231–275.
- R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
- R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
- C. R. Parker and P. J. Bones, “Convergence of iterative phase retrieval improved by utilizing zero sheets,” Opt. Commun. 92, 209–214 (1992).
- J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
- J. H. Seldin and J. R. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7, 412–427 (1990).
- H. Takajo, T. Takahashi, H. Kawanami, and 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).
- H. Takajo, T. Takahashi, R. Ueda, and 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).
- H. Takajo, T. Takahashi, and T. 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).
- We use this simplification throughout this paper. That is, the term “the support” always means “the support imposed as the object-domain constraint.”
- R. G. Lane, “Phase retrieval using conjugate gradient minimization,” J. Modern Opt. 38, 1797–1813 (1991).
- H. Takajo, T. Shizuma, T. Takahashi, and S. Takahata, “Reconstruction of an object from its noisy Fourier modulus: Ideal estimate of the object to be reconstructed and a method that attempts to find that estimate,” Appl. Opt. 38, 5568–5576 (1999).
- A. Levi and H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
- A. Levi and H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, Calif., 1987), Chap. 8, pp. 277–320.
- Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
- M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
- The CPHIO can converge to one of the output-stagnation objects of the HIO unless the value of β is too large. 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 is 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.
- As the size of the object to be reconstructed becomes large, the number of the output-stagnation objects increases dramatically and, in addition, the way the territories of the output-stagnation objects are formed in the MN-dimensional space becomes complicated. So, strictly speaking, Fig. 3(a) does not show that the HIO/ER and, thus, Eom fell into a ‘completely’ periodic state as in the case of the 2 × 2 objects with L-shaped support, which was discussed in Ref. 10. Figure 3(a) does, however, allow us to say that Eom fell into an ‘effectively’ periodic state.

## 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.