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 16, 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 15, 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

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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. 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. R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
    [CrossRef]
  5. C. R. Parker, P. J. Bones, “Convergence of iterative phase retrieval improved by utilizing zero sheets,” Opt. Commun. 92, 209–214 (1992).
    [CrossRef]
  6. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  7. J. H. Seldin, J. R. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7, 412–427 (1990).
    [CrossRef]
  8. 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]
  9. 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]
  10. H. Takajo, T. Takahashi, 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).
    [CrossRef]
  11. We use this simplification throughout this paper. That is, the term “the support” always means “the support imposed as the object-domain constraint.”
  12. R. G. Lane, “Phase retrieval using conjugate gradient minimization,” J. Modern Opt. 38, 1797–1813 (1991).
    [CrossRef]
  13. H. Takajo, T. Shizuma, T. Takahashi, 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).
    [CrossRef]
  14. 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]
  15. 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.
  16. Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  17. 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).
    [CrossRef]
  18. 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.
  19. 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.

1999 (2)

1998 (1)

1997 (1)

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)

R. G. Lane, “Phase retrieval using conjugate gradient minimization,” J. Modern Opt. 38, 1797–1813 (1991).
[CrossRef]

1990 (1)

1987 (1)

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

1986 (1)

1984 (1)

1982 (2)

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).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

1979 (1)

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

1972 (1)

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).

Bones, P. J.

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

Bruck, Y. M.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

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.

Fienup, J. R.

Gerchberg, R. W.

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).

Hayes, M. H.

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).
[CrossRef]

Kawanami, H.

Lane, R. G.

R. G. Lane, “Phase retrieval using conjugate gradient minimization,” J. Modern Opt. 38, 1797–1813 (1991).
[CrossRef]

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

Levi, A.

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]

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.

Parker, C. R.

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

Saxton, W. O.

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).

Seldin, J. H.

Shizuma, T.

Sodin, L. G.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Stark, H.

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]

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.

Takahata, S.

Takajo, H.

Taninaka, M.

Ueda, R.

Wackerman, C. C.

Appl. Opt. (2)

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

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).
[CrossRef]

J. Modern Opt. (1)

R. G. Lane, “Phase retrieval using conjugate gradient minimization,” J. Modern Opt. 38, 1797–1813 (1991).
[CrossRef]

J. Opt. Soc. Am. A (6)

Opt. Commun. (3)

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

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

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Optik (1)

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).

Other (5)

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.

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.

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.

We use this simplification throughout this paper. That is, the term “the support” always means “the support imposed as the object-domain constraint.”

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Figures (9)

Fig. 1
Fig. 1

Schematic representation of how the HIO behaves in the output-stagnating state in which the output object of the HIO stays at a convergence output object of the ER, denoted as gs,ER and how the HIO emerges from it. P and Q represent the points gs,ER D and gs,ER, respectively. In this figure, the space passing through point P and parallel to the complement of the ODC space is assumed to be of two dimensions and to coincide with the plane of the paper. In this output-stagnating state, although the output object of the HIO stays at point Q, the input object moves along the emerging path, which lies in the plane of the paper and is parallel to the vector gs,ER ND. In this figure, three examples of the emerging path are indicated.

Fig. 2
Fig. 2

Original object.

Fig. 3
Fig. 3

Relations of E om versus the number of iterations m in the uncontaminated case. (a), Relation obtained by the HIO/ER; (b) relations obtained by the HIO/CPHIO (70/230), by the HIO/CPHIO (70/930), by the HIO alone.

Fig. 4
Fig. 4

Reconstructed objects in the uncontaminated case. (a) Reconstructed object by 3 × 104 iterations of the HIO/ER, for which E om = 1.74 × 10-2 and E tm = 3.48 × 10-1. (b) Reconstructed object at the 9 × 103th iteration of the HIO/CPHIO (70/230), for which E om = 8.63 × 10-6 and E tm = 9.62 × 10-6.

Fig. 5
Fig. 5

Relations of E tm versus N R : (open circle) for the reconstructed objects by 100 cycles of the HIO/CPHIO (70/230); (filled circle) for the reconstructed objects by 300 cycles of the HIO/CPHIO (70/930); (open triangle) for the reconstructed objects by 3 × 104 iterations of the HIO alone.

Fig. 6
Fig. 6

Reconstructed objects in the case in which N R = 8%. (a) Reconstructed object by 300 cycles of the HIO/CPHIO (70/930), for which E om = 3.71 × 10-2 and E tm = 1.84 × 10-1. (b) Reconstructed object by 3 × 104 iterations of the HIO alone, for which E om = 7.10 × 10-2, E tm = 2.28 × 10-1.

Fig. 7
Fig. 7

Reconstructed objects in the case in which N R = 15%. (a) Reconstructed object by 300 cycles of the HIO/CPHIO (70/930), for which E om = 5.62 × 10-2 and E tm = 3.01 × 10-1. (b) Reconstructed object by 3 × 104 iterations of the HIO alone, for which E om = 1.03 × 10-1 and E tm = 4.00 × 10-1.

Fig. 8
Fig. 8

Relations of E tm versus N R : (filled triangle) for the reconstructed objects by 6 × 104 iterations of the SHIO preceded by 3 × 104 iterations of the HIO: (open circle) for the reconstructed objects by 6 × 104 iterations of the SHIO preceded by 300 cycles of the HIO/CPHIO (70/930). In this figure, the relation for the reconstructed objects by 300 cycles of the HIO/CPHIO (70/930) is also indicated by use of the (filled circle) symbol, although this relation has already been presented in Fig. 5.

Fig. 9
Fig. 9

Reconstructed object by 6 × 104 iterations of the SHIO preceded by 300 cycles of the HIO/CPHIO (70/930), for which E om = 5.22 × 10-2 and E tm = 2.81 × 10-1; N R = 15%.

Equations (34)

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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, vexpi2πux/M+vy/N,
gm+1x, y=gmx, yx, yD0x, yD,
Fm2=MN-1uv|Gmu, v|-|Fu, v|2,
om2=x,yD |gmx, y|2,
Fm2=xy |gmx, y-gmx, y|2.
F2=xy |gx, y-gx, y|2.
F2gx, y=2gx, y-gx, y.
F2gx, yg=gm=2gmx, y-gmx, y
gm+1x, y=gmx, y-12F2gx, yg=gmx, yD0x, yD.
gm+1x, y=gmx, y.
F2gx, yg=gm=0 x, yD
gmx, y=gmx, y x, yD
gmx, y=gmx, y.
Fm2om2F,m+12o,m+12.
gm+1x, y=gmx, yx, yDgmx, y-βgmx, yx, yD,
Gm+1u, v=Gmu, v+DFT-12F2gx, yg=gmD-iβ MN4|Fu, v|2o2θu, vG=Gm×Gmu, v-β MN4|Fu, v|o2|Gu, v|G=Gm×Gmu, v,
-12F2gx, yg=gmD=-12F2gx, yg=gmx, yD0x, yD
Gm+1u, v=Gmu, v+DFT-12F2gx, yg=gmD-iβ MN4|Fu, v|2o2θu, vG=GmGmu, v,
Gm+1u, v=GmNDu, v+GmDu, v+iβ 1|Fu, v|2ImGmu, v×GmNDu, v*Gmu, v,
gmNDx, y=0x, yDgmx, yx, yD,
F2gx, yg=gm =2gmx, y-gmx, y =0, x, yD,
o2θu, vG=Gm=-4MNImGmu, vGmNDu, v*=0.
Gm+ku, v=Gmu, v-kβGsNDu, v
|Gmu0, v0|<k0β|GsNDu0, v0|
|Gm-u0, -v0|<k0β|GsND-u0, -v0|,
Eom=om2MN-1uv |Fu, v|21/2
Etm=x,yS |g˜mx, y-fx, y|2xy |fx, y|21/2,
NR=uv|Fnu, v|-|Fu, v|2uv |Fu, v|21/2×100%,
x, y|x, ySx, y|x, yD
gm+1NSx, y=gmNSx, y-βgmNSx, y
gmNSx, y=0x, ySgmx, yx, yS,
gm+1NSx, y=g0NSx, y-β k=0m gkNSx, y.
gmNSx, y=gnNSx, y

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