## Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison

Applied Optics, Vol. 33, Issue 2, pp. 209-218 (1994)

http://dx.doi.org/10.1364/AO.33.000209

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### Abstract

A detailed comparison of the original Gerchberg–Saxton and the Yang–Gu algorithms for the reconstruction of model images from two intensity measurements in a nonunitary transform system is presented. The Yang–Gu algorithm is a generalization of the Gerchberg–Saxton algorithm and is effective in solving the general amplitude–phase-retrieval problem in any linear unitary or nonunitary transform system. For a unitary transform system the Yang–Gu algorithm is identical to the Gerchberg–Saxton algorithm. The reconstruction of images from data corrupted with random noise is also investigated. The simulation results show that the Yang–Gu algorithm is relatively insensitive to the presence of noise in data. In all cases studied the Yang–Gu algorithm always resulted in a highly accurate recovered phase.

© 1994 Optical Society of America

**History**

Original Manuscript: September 8, 1992

Revised Manuscript: July 15, 1993

Published: January 10, 1994

**Citation**

Guo-zhen Yang, Bi-zhen Dong, Ben-yuan Gu, Jie-yao Zhuang, and Okan K. Ersoy, "Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison," Appl. Opt. **33**, 209-218 (1994)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-33-2-209

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### References

- H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, New York, 1978), pp. 13–19. [CrossRef]
- E. L. O'Neill, A. Walther, “The question of phase in image formation,” Opt. Acta 10, 33–40 (1963). [CrossRef]
- B. J. Hoenders, “On the solution of the phase retrieval problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).
- R. W. Gerchberg, W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).
- R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 227–246 (1972).
- W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
- W. O. Saxton, “Recovery of specimen information for strongly scattering objects,” in Computer Processing of Electron Microscope Image, P. W. Hawkes, ed. (Springer-Verlag, Berlin, 1980), pp. 35–87. [CrossRef]
- R. W. Gerchberg, W. O. Saxton, “Wave phase from image and diffraction plane pictures,” in Image Processing and Computer-Aided Design in Electron Optics, P. W. Hawkes, ed. (Academic, New York, 1973), pp. 66–81.
- D. L. Misell, “The phase problem in electron microscopy,” in Advances in Optical and Electron Microscopy, V. E. Cosslett, R. Barber, eds. (Academic, London, 1978), Vol. 7, pp. 185–276.
- H. A. Ferwerda, B. J. Hoenders, A. M. J. Huiser, P. Van Toorn, “On the phase reconstruction problem in light and electron microscopy,” Photogr. Sc. Eng. 21, 282–289 (1977).
- I. LaHaie, “Inverse source problem for three-dimensional partially coherent sources and fields,” J. Opt. Soc. Am. A 2, 35–44(1985). [CrossRef]
- R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990). [CrossRef]
- R. W. Gerchberg, W. O. Saxton, “Comment on ‘A method for the solution of the phase problem in electron microscopy,’” J. Phys. D 6, L31–L32 (1973);“A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972). [CrossRef]
- D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973). [CrossRef]
- C. H. Slump, H. A. Ferwerda, “Statistical analysis of low-dose reconstruction of weak phase-amplitude objects from two defocused images. I.,” Optik 62, 93–104 (1982).
- A. M. J. Huiser, P. Van Toorn, H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern. III. The development of an algorithm,” Optik 47, 1–8 (1977).
- P. Van Toorn, H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern. IV. Checking of algorithm by means of simulated objects,” Optik 47, 123–134 (1977).
- R. H. Boucher, “Convergence of algorithm for phase retrieval from two intensity distributions,” in 1980 International Optical Computing Conference I, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 130–141 (1980).
- J. R. Fenup, T. R. Crimmins, W. Holsztynski, “Reconstruction of the support of an object from the support of its autocorrelation,” J. Opt. Soc. Am. 72, 610–624 (1982). [CrossRef]
- T. R. Crimmins, J. R. Fienup, “Uniqueness of phase retrieval for functions with sufficiently disconnected support,” J. Opt. Soc. Am. 73, 218–221 (1983). [CrossRef]
- T. R. Crimmins, “Phase retrieval for discrete functions with support constraints,” J. Opt. Soc. Am. A 4, 124–134 (1987). [CrossRef]
- 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]
- J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
- J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
- R. W. Gerchberg, “Superresolution through error energy reduction,” Opt. Acta 21, 709–720 (1974). [CrossRef]
- R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase-restoration procedure,” J. Opt. Soc. Am. 73, 358–365 (1983). [CrossRef]
- A. Levi, 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). [CrossRef]
- R. Barakat, G. Newsam, “Algorithm for reconstruction of partially known, band-limited Fourier-transform pairs from noisy data,” J. Opt. Soc. Am. A 2, 2027–3039 (1985). [CrossRef]
- B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).
- G. Yang, B. Gu, “On the amplitude-phase retrieval problem in the optical system,” Acta Phys. Sin. 30, 410–413 (1981).
- G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude-phase retrieval problem in an optical system involved nonunitary transformation,” Optik 75, 68–74 (1987).
- B. Gu, G. Yang, B. Dong, “General theory for performing an optical transform,” Appl. Opt. 25, 3197–3206 (1986). [CrossRef] [PubMed]
- E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1986).
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
- G. Yang, B. Gu, B. Dong, “Theory of the amplitude-phase retrieval in any linear transform system and its applications,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1767, 457–479 (1992).
- J. Ximen, J. Yan, “Computational trials for phase retrieval in electron microscopy from image and diffraction patterns,” Acta Phys. Sin. 32, 762–769 (1983).

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