Abstract

In general, the problem of reconstructing an object from its Fourier modulus has no solution when the Fourier modulus is contaminated by noise. Therefore a quasi solution, which we call the ideal estimate of the object to be reconstructed, is defined here based on the concept of territories of the convergence objects of the error-reduction algorithm, and a method that attempts to find that solution is presented. Keeping in mind that the ideal estimate is one of the output-stagnation objects of the hybrid input–output algorithm, we modify the hybrid input–output algorithm so that the output-stagnation objects can be located even when the value of the feedback parameter is not infinitesimally small, and this modified algorithm is combined with the hybrid input–output algorithm itself. The results of computer simulations carried out to test the performance of the proposed method are shown.

© 1999 Optical Society of America

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References

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  1. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  2. 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, Calif., 1987), Chap. 7, pp. 231–275.
  3. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  4. J. H. Seldin, J. R. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7, 412–427 (1990).
    [CrossRef]
  5. 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]
  6. 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]
  7. 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]
  8. 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]
  9. 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]
  10. C. R. Parker, P. J. Bones, “Convergence of iterative phase retrieval improved by utilizing zero sheets,” Opt. Commun. 92, 209–214 (1992).
    [CrossRef]
  11. 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]
  12. A. Levi, 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.
  13. R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
    [CrossRef]
  14. 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]
  15. A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problem (Wiley, New York, 1977).
  16. 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]
  17. Since the error-reduction algorithm can always converge, the error-reduction algorithm can relate any object in the input-object space, which is formed by the set of objects satisfying the object-domain constraints, to some convergence object in the sense that, if the error-reduction algorithm starts from the object, it reaches the convergence object. The territory of a convergence object was defined as the subspace in the input-object space that is formed by the set of objects related to the convergence object in this sense.
  18. B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
    [CrossRef]

1998 (1)

1997 (1)

1996 (1)

1995 (1)

1994 (2)

1992 (1)

C. R. Parker, P. J. Bones, “Convergence of iterative phase retrieval improved by utilizing zero sheets,” Opt. Commun. 92, 209–214 (1992).
[CrossRef]

1991 (1)

1990 (1)

1989 (1)

B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

1987 (2)

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]

R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
[CrossRef]

1986 (1)

1982 (1)

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problem (Wiley, New York, 1977).

Bates, R. H. T.

B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

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]

Bones, P. J.

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]

C. R. Parker, P. J. Bones, “Convergence of iterative phase retrieval improved by utilizing zero sheets,” Opt. Commun. 92, 209–214 (1992).
[CrossRef]

Chen, P.-T.

Dainty, J. C.

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, Calif., 1987), Chap. 7, pp. 231–275.

Fiddy, M. A.

Fienup, J. R.

Fright, W. R.

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]

Kawanami, H.

Lane, R. G.

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]

R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
[CrossRef]

Levi, A.

A. Levi, 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.

Liao, C.-W.

McCallum, B. C.

B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

Parker, C. R.

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]

C. R. Parker, P. J. Bones, “Convergence of iterative phase retrieval improved by utilizing zero sheets,” Opt. Commun. 92, 209–214 (1992).
[CrossRef]

Pommet, D. A.

Satherley, B. L.

Seldin, J. H.

Stark, H.

A. Levi, 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.

Takahashi, T.

Takajo, H.

Taninaka, M.

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problem (Wiley, New York, 1977).

Ueda, R.

Wackerman, C. C.

Watson, R. W.

Yagle, A. E.

Appl. Opt. (1)

IEEE Trans. Acoust. Speech Signal Process. (1)

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]

J. Mod. Opt. (1)

B. C. McCallum, R. H. T. Bates, “Towards a strategy for automatic phase retrieval from noisy Fourier intensities,” J. Mod. Opt. 36, 619–648 (1989).
[CrossRef]

J. Opt. Soc. Am. A (9)

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]

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]

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]

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]

J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
[CrossRef]

J. H. Seldin, J. R. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7, 412–427 (1990).
[CrossRef]

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]

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]

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]

Opt. Commun. (2)

C. R. Parker, P. J. Bones, “Convergence of iterative phase retrieval improved by utilizing zero sheets,” Opt. Commun. 92, 209–214 (1992).
[CrossRef]

R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
[CrossRef]

Other (4)

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problem (Wiley, New York, 1977).

Since the error-reduction algorithm can always converge, the error-reduction algorithm can relate any object in the input-object space, which is formed by the set of objects satisfying the object-domain constraints, to some convergence object in the sense that, if the error-reduction algorithm starts from the object, it reaches the convergence object. The territory of a convergence object was defined as the subspace in the input-object space that is formed by the set of objects related to the convergence object in this sense.

A. Levi, 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.

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, Calif., 1987), Chap. 7, pp. 231–275.

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Figures (6)

Fig. 1
Fig. 1

Original object.

Fig. 2
Fig. 2

Relation of E om versus the number of iterations m obtained by the HIO-SHIO in the case in which N R = 1.19%. (a) Relation in the step in which the HIO was executed. (b) Relation in the step in which the SHIO was executed.

Fig. 3
Fig. 3

Reconstructed objects in the case in which N R = 1.19%. (a) Reconstructed object by the HIO-SHIO, for which E om = 9.284 × 10-3 and E tm = 1.625 × 10-2. (b) Reconstructed object in the HIO step, for which E om = 1.631 × 10-2 and E tm = 3.556 × 10-2. (c) The ideal estimate, for which E om = 9.283 × 10-3 and E tm = 1.621 × 10-2.

Fig. 4
Fig. 4

Reconstructed objects in the case in which N R = 5.88%. (a) Reconstructed object by the HIO-SHIO, for which E om = 4.514 × 10-2 and E tm = 1.481 × 10-1. (b) Reconstructed object in the HIO step, for which E om = 9.350 × 10-2 and E tm = 3.625 × 10-1. (c) The ideal estimate, for which E om = 4.362 × 10-2 and E tm = 8.790 × 10-2.

Fig. 5
Fig. 5

Relations of E tm versus N R for the reconstructed objects by the HIO- SHIO, the reconstructed objects in the HIO step, and for the ideal estimates: (●) reconstructed objects by the HIO-SHIO, (△) reconstructed objects in the HIO step, (○) ideal estimates.

Fig. 6
Fig. 6

Relations of E om versus N R for the reconstructed objects by the HIO-SHIO, the reconstructed objects in the HIO step, and for the ideal estimates: (●) reconstructed objects by the HIO-SHIO, (△) reconstructed objects in the HIO step, (○) ideal estimates.

Equations (21)

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Fu, v=|Fu, v|expiϕu, v=x=-M/2+1M/2y=-N/2+1N/2 fx, y×exp-i2πux/M+vy/N,
fx, y=MN-1u=-M/2+1M/2v=-N/2+1N/2 Fu, v×expi2πux/M+vy/N,
gm+1x, y=gmx, y,x, yD0,x, yD,
gm+1x, y=gmx, y,x, yDgmx, y-βgmx, y,x, yD,
Fm2=MN-1uv|Gmu, v|-|Fu, v|2,
om2=x,yD |gmx, y|2,
NR=uv|Fnu, v|-|Fu, v|2uv |Fu, v|21/2×100%,
Gm+1u, v=Gmu, v+DFT-12F2gx, yg=gmD-iβ MN4|Fu, v|2o2θu, vG=GmGmu, v-β MN4|Fu, v|o2|Gu, v|G=GmGmu, v,
-12F2gx, yg=gmD=-12F2gx, yg=gm,x, yD0,x, yD
Gm+1u, v=Gmu, v+DFT-12F2gx, yg=gmD-iβ MN4|Fu, v|2o2θu, vG=Gm Gmu, v,
Gm+1u, v=GmNDu, v+GmDu, v+iβ 1|Fu, v|2 ImGmu, v×GmNDu, v*Gmu, v,
gmNDx, y=0,x, yDgmx, y,x, yD,
F2gx, yg=gm=2gmx, y-gmx, y=0,  x, yD,
o2θu, vG=Gm=-4MN ImGmu, v×GmNDu, v*=0,
Gm+ku, v=Gmu, v-kβGsNDu, v
|Gmu0, v0|<k0β|GsNDu0, v0|,
ru0, v0=|Gmu0, v0||GsNDu0, v0|.
Gm+1u, v=Gmu, v-r1+r22 GsNDu, v.
Tm=uvImGmu, vGmNDu, v*2uv |Gmu, v|2uv |GmNDu, v|21/2,
Eom=om2MN-1uv |Fu, v|21/2
Etm=xy |g˜mx, y-fx, y|2xy |fx, y|21/2,

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