Abstract

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

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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. 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.

1998

1997

1992

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

1990

1987

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

1986

1982

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]

Fienup, J. R.

Kawanami, H.

Lane, R. G.

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

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]

Seldin, J. H.

Takahashi, T.

Takajo, H.

Taninaka, M.

Ueda, R.

Wackerman, C. C.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

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]

Other

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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Figures (4)

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. Characters A and B represent the points gs,ERS and gs,ER, respectively, and the symbol g˜s,ERNS represents the vector originating from point A and terminating at point B. In this figure the space passing through point A and being parallel to the complement of the SC 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 B, the input object moves along the emerging path, which lies in the plane of the paper and is parallel to the vector g˜s,ERNS. In this figure three examples of the emerging path are indicated.

Fig. 2
Fig. 2

Eom and Wm (× 0.1) versus the number of iterations m obtained by the HIO with β=0.1. The original object f is specified by Eq. (12). The initial input object gs2 is given by Eq. (14). As the object-domain constraint, the support constraint was imposed.

Fig. 3
Fig. 3

Values of gm(1, 0) and gm(5, 3) versus the number of iterations m in the experiment shown in Fig. 2: results for (a) gm(1, 0) and (b) gm(5, 3).

Fig. 4
Fig. 4

Eom and Wm (× 0.1) versus the number of iterations m obtained by the HIO/ER whose one cycle was composed of 90 iterations of the HIO with β=0.7 and 10 iterations of the ER. The original object is f. The initial input object is gs2. As the object-domain constraint, the support constraint was imposed.

Equations (17)

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F(u, v)= |F(u, v)|exp[iϕ(u, v)]=x=0M-1y=0N-1f(x, y)exp[-i2π(ux/M+vy/N)].
gm+1(x, y)
=gm(x, y)(x, y)Dgm(x, y)-βgm(x, y)(x, y)D, 
gm+1(x, y)=gm(x, y)(x, y)D0(x, y)D.
Fm2=(MN)-1uv[|Gm(u, v)|-|F(u, v)|]2
om2=(x, y)D|gm(x, y)|2,
gm+1(x, y)=gm(x, y)(x, y)Sgm(x, y)-βgm(x, y)(x, y)S,
gm+1(x, y)=gm(x, y)(x, y)S0(x, y)S,
gm+1NS(x, y)=gmNS(x, y)-βgmNS(x, y),
gmNS(x, y)=0,(x, y)Sgm(x, y),(x, y)S .
gm+1NS(x, y)=g0NS(x, y)-β k=0mgkNS(x, y).
gmNS(x, y)=gnNS(x, y)
f=0.4763.2441.3791.6592.9391.102,
gs1=0.3532.1433.1720.6842.4701.976
gs2=0.2661.8762.9710.7462.7112.222.
Eom=om2(MN)-1uv|F(u, v)|21/2,
Wm=xy[gmNS(x, y)]21/2,

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