Abstract

Even though the hybrid input–output algorithm (HIO) has been recognized empirically to be one of the most successful versions of the iterative Fourier transform algorithm used for phase retrieval, its behavior is not yet well understood. Therefore a theoretical investigation on the convergence property of the HIO with an infinitesimally small feedback parameter is presented, although on a rather intuitive level, and it is shown that, until a solution is found, this algorithm continues to travel among the objects seeking those that satisfy the Fourier-domain constraint and for which the object-domain error has a locally minimum value. The concept of the territory is introduced with use of the algorithm constructed by modifying the HIO, and then the results are presented of the computer simulations for 2×2 objects with L-shaped support that were carried out to test the validity of our theory and to gain insight into the case in which the value of the feedback parameter is finite.

© 1998 Optical Society of America

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References

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  1. 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, 1987).
  2. H. H. Arsenault, K. Chalasinska-Macukow, “The solution of the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
    [CrossRef]
  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. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  5. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  6. J. H. Seldin, J. R. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7, 412–427 (1990).
    [CrossRef]
  7. 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]
  8. 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]
  9. 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]
  10. 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]
  11. 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]
  12. C. R. Parker, P. J. Bones, “Convergence of iterative phase retrieval improved by utilizing zero sheets,” Opt. Commun. 92, 209–214 (1992).
    [CrossRef]
  13. 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]
  14. 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]
  15. R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
    [CrossRef]
  16. H. Takajo, T. Takahashi, “Competence of hybrid input–output algorithm for phase retrieval,” Jpn. J. Opt. 22, 419–427 (1993).

1997

1996

1995

1994

1993

H. Takajo, T. Takahashi, “Competence of hybrid input–output algorithm for phase retrieval,” Jpn. J. Opt. 22, 419–427 (1993).

1992

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

1991

1990

1987

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

1986

1983

H. H. Arsenault, K. Chalasinska-Macukow, “The solution of the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
[CrossRef]

1982

1972

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).

Arsenault, H. H.

H. H. Arsenault, K. Chalasinska-Macukow, “The solution of the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
[CrossRef]

Bates, R. H. T.

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]

Chalasinska-Macukow, K.

H. H. Arsenault, K. Chalasinska-Macukow, “The solution of the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
[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, 1987).

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]

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).

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]

Liao, C.-W.

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.

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.

Takahashi, T.

Takajo, H.

Ueda, R.

Wackerman, C. C.

Watson, R. W.

Yagle, A. E.

Appl. Opt.

IEEE Trans. Acoust. Speech Signal Process.

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. Opt. Soc. Am. A

Jpn. J. Opt.

H. Takajo, T. Takahashi, “Competence of hybrid input–output algorithm for phase retrieval,” Jpn. J. Opt. 22, 419–427 (1993).

Opt. Commun.

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]

H. H. Arsenault, K. Chalasinska-Macukow, “The solution of the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
[CrossRef]

Optik

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

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, 1987).

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Figures (11)

Fig. 1
Fig. 1

Geometrical representation of each term in Eq. (27) in the complex plane. In this figure the second and third terms on the right-hand side of Eq. (27) are designated A and B, respectively.

Fig. 2
Fig. 2

Schematic representation of the behavior of the IHIO in the MN-dimensional space.

Fig. 3
Fig. 3

Behavior of the CPHIO in the case in which the original object f=(1, 1.5, 3.5) and β=0.01. Open circles represent the positions of the initial input objects.

Fig. 4
Fig. 4

Behavior of the HIO in the case in which f=(1, 1.5, 3.5) and β=0.01. The g0 depicted in Fig. 3 is used as the initial input object.

Fig. 5
Fig. 5

Eom versus the number of iterations m in the experiment shown in Fig. 4.

Fig. 6
Fig. 6

Sm or Wm versus the number of iterations m in the experiment shown in Fig. 4.

Fig. 7
Fig. 7

Behavior of the HIO in the case in which f=(1, 1.5, 3.5) and the initial input object is the g0 depicted in Fig. 3. (a) β=0.1, (b) β=1.

Fig. 8
Fig. 8

Eom versus the number of iterations m in the experiment shown in Fig. 7.

Fig. 9
Fig. 9

Number of iterations Nr required for decreasing Eom to 10-4 versus β. The original object is f1=(1, 1.5, 3.5), f2=(1, 1.5, 10.5), or f3=(1, 1.5, 20.5) and the initial input object is the g0 depicted in Fig. 3.

Fig. 10
Fig. 10

Behavior of the CPHIO in the case in which f=(1, 1.8, 5) and β=0.01.

Fig. 11
Fig. 11

Behavior of the HIO for a hole. The initial input object is (1, 1, 1), and f=(1, 1.8, 5).

Equations (59)

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F(u, v)=|F(u, v)|exp[iϕ(u, v)]=x=-M/2+1M/2y=-N/2+1N/2f(x, y)×exp[-i2π(ux/M+vy/N)],
f(x, y)=(MN)-1u=-M/2+1M/2v=-N/2+1N/2F(u, v)×exp[i2π(ux/M+vy/N)],
gm+1(x, y)=gm(x, y)(x, y)D0(x, y)D,
Fm2=(MN)-1uv[|Gm(u, v)|-|F(u, v)|]2=(MN)-1uv|Gm(u, v)-Gm(u, v)|2
om2=(x, y)D|gm(x, y)|2,
Fm2=xy|gm(x, y)-gm(x, y)|2.
F2=xy|g(x, y)-g(x, y)|2.
(F2)g(x, y)=2[g(x, y)-g(x, y)].
(F2)g(x, y)g=gm=2[gm(x, y)-gm(x, y)]
gm+1(x, y)
=gm(x, y)-12(F2)g(x, y)g=gm,(x, y)D0(x, y)D.
gm+1(x, y)=gm(x, y).
(F2)g(x, y)g=gm=0(x, y)D
gm(x, y)=gm(x, y)(x, y)D
gm(x, y)=gm(x, y).
Fm2om2F,m+12o,m+12,
gm+1(x, y)=gm(x, y)(x, y)Dgm(x, y)-βgm(x, y)(x, y)D,
gm(x, y)=gm(x, y)-βgm(x, y)(x, y)D.
gm(x, y)=0(x, y)D.
gm+1(x, y)=gm(x, y)+-12(F2)g(x, y)g=gm0-β0(x, y)Dgm(x, y)(x, y)D
gm+1(x, y)=gm(x, y)+-12(F2)g(x, y)g=gm(x, y)D0(x, y)D,
(F2)g(x, y)g=gm=2[gm(x, y)-gm(x, y)]=0  (x, y)D.
gm+1(x, y)=gm(x, y)-β0(x, y)Dgm(x, y)(x, y)D=gm(x, y)-βgmND(x, y)
02=(x, y)D|g(x, y)|2.
GmND(u, v)=ip(o2)θ(u, v)G=GmGm(u, v)+q(o2)|G(u, v)|G=GmGm(u, v),
p=MN4|F(u, v)|2
q=MN4|F(u, v)|,
Gm+1(u, v)=Gm(u, v)-iβp(o2)θ(u, v)G=GmGm(u, v)-βq(o2)|G(u, v)|G=GmGm(u, v).
Gm+1(u, v)=Gm(u, v)-iβp(o2)θ(u, v)G=GmGm(u, v).
(o2)θ(u, v)G=Gm=0
Gm+1(u, v)=Gm(u, v)-βq(o2)|G(u, v)|G=GmGm(u, v)
(o2)θ(u, v)G=Gm=-4MNIm[Gm(u, v){GmND(u, v)}*].
Im[Gm(u, v){GmND(u, v)}*]=0,
phase{Gm(u, v)}=phase{GmND(u, v)}(modπ)
Gm+1(u, v)=Gm(u, v)-βq(o2)|G(u, v)|G=GsGs(u, v),
q(o2)|G(u, v)|G=GsGs(u, v)=GsND(u, v)
Gm+1(u, v)=Gm(u, v)-βGsND(u, v).
Gm+k(u, v)=Gm(u, v)-kβGsND(u, v).
uv|GsD(u, v)|2=uv|Gs(u, v)-GsND(u, v)|2=uv[|Gs(u, v)|+|GsND(u, v)|]2>uv|Gs(u, v)|2
|Gm(u0, v0)|<k0β|GsND(u0, v0)|,
Gm+1(u, v)=Gm(u, v)+DFT-12(F2)g(x, y)g=gmD-iβp(o2)θ(u, v)G=GmGm(u, v)-βq(o2)|G(u, v)|G=GmGm(u, v).
Gm+1(u, v)=Gm(u, v)+DFT-12(F2)g(x, y)g=gmD-iβp(o2)θ(u, v)G=GmGm(u, v).
Gm+1(u, v)=GmND(u, v)+GmD(u, v)+iβ 1|F(u, v)|2Im[Gm(u, v)×{GmND(u, v)}*]Gm(u, v).
Eom=om2(MN)-1uv|F(u, v)|21/2
Sm=xy[gmD(x, y)-gmD(x, y)]21/2
Wm=xy[gmND(x, y)]21/2,
(o2)θ(u, v)=2(x, y)Dg(x, y) g(x, y)θ(u, v).
g(x, y)θ(u, v)=(MN)-1 θ(u, v)uv|G(u, v)|×exp[iθ(u, v)+i2π(ux/M+vy/N)]=i(MN)-1|G(u, v)|exp[iθ(u, v)+i2π(ux/M+vy/N)]-i(MN)-1|G(u, v)|exp[-iθ(u, v)-i2π(ux/M+vy/N)]=i(MN)-1G(u, v)exp[i2π(ux/M+vy/N)]-i(MN)-1G*(u, v)exp[-i2π(ux/M+vy/N)].
(o2)θ(u, v)=i2(MN)-1G(u, v)(x, y)Dg(x, y)×exp[i2π(ux/M+vy/N)]-i2(MN)-1G*(u, v)(x, y)Dg(x, y)×exp[-i2π(ux/M+vy/N)]=i2(MN)-1G(u, v){GND(u, v)}*-i2(MN)-1{G(u, v)}*GND(u, v)=-4(MN)-1 Im[G(u, v){GND(u, v)}*].
(o2)θ(u, v)G=Gm=-4(MN)-1 Im[Gm(u, v)×{GmND(u, v)}*]
(o2)|G(u, v)|=2(x, y)Dg(x, y) g(x, y)|G(u, v)|.
(o2)|G(u, v)|=2(MN)-1{GND(u, v)}* exp[iθ(u, v)]+2(MN)-1GND(u, v)exp[-iθ(u, v)]=4(MN)-1|G(u, v)|-1 ×Re[G(u, v){GND(u, v)}*].
(o2)|G(u, v)|G=Gm=4(MN)-1|F(u, v)|-1 ×Re[Gm(u, v){GmND(u, v)}*]
Gm(u, v){GmND(u, v)}*
=MN4-i(o2)θ(u, v)G=Gm+|F(u, v)|(o2)|G(u, v)|G=Gm
GmND(u, v)=i MN4|F(u, v)|2(o2)θ(u, v)G=GmGm(u, v)+MN4|F(u, v)|(o2)|G(u, v)|G=Gm×Gm(u, v)
gmD(x, y)=gmD(x, y).
gmND(x, y)=gm(x, y)-gmD(x, y).
gmND(x, y)=gm(x, y)-gm(x, y).

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