Abstract

This study sheds new light on the stagnation problem troubling the iterative Fourier transform algorithm used for phase retrieval. This is implemented by the introduction of two concepts, the territory and the hole, which exist in a relationship in the sense that the creation and the annihilation of a hole results in the annihilation and the creation of a corresponding territory. These concepts are brought about in the parameter space formed by the set of objects satisfying the object-domain constraints by means of topographical investigations of the minimization process by use of the error-reduction algorithm, paying particular attention to the cases of 2×2 and 3×2 objects with L-shaped supports. The effect of noise contained in the Fourier modulus on the stagnation phenomena is also clarified by use of these concepts.

© 1997 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. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  3. 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]
  4. A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, 1987), Chap. 8, pp. 277–320.
  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. J. H. Seldin, J. R. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7, 412–427 (1990).
    [CrossRef]
  13. T. Takahashi, H. Takajo, S. Mattori, N. Nagano, “Numerical investigation of the stagnation problems of an iterative phase-retrieval algorithm,” Jpn. J. Opt. 21, 119–127 (1992).

1996

1995

1994

1992

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

T. Takahashi, H. Takajo, S. Mattori, N. Nagano, “Numerical investigation of the stagnation problems of an iterative phase-retrieval algorithm,” Jpn. J. Opt. 21, 119–127 (1992).

1991

1990

1987

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

1984

1982

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]

Chen, P.-T.

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]

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]

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, 1987), Chap. 8, pp. 277–320.

Liao, C.-W.

Mattori, S.

T. Takahashi, H. Takajo, S. Mattori, N. Nagano, “Numerical investigation of the stagnation problems of an iterative phase-retrieval algorithm,” Jpn. J. Opt. 21, 119–127 (1992).

Nagano, N.

T. Takahashi, H. Takajo, S. Mattori, N. Nagano, “Numerical investigation of the stagnation problems of an iterative phase-retrieval algorithm,” Jpn. J. Opt. 21, 119–127 (1992).

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, “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, 1987), Chap. 8, pp. 277–320.

Takahashi, T.

T. Takahashi, H. Takajo, S. Mattori, N. Nagano, “Numerical investigation of the stagnation problems of an iterative phase-retrieval algorithm,” Jpn. J. Opt. 21, 119–127 (1992).

Takajo, H.

T. Takahashi, H. Takajo, S. Mattori, N. Nagano, “Numerical investigation of the stagnation problems of an iterative phase-retrieval algorithm,” Jpn. J. Opt. 21, 119–127 (1992).

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.

T. Takahashi, H. Takajo, S. Mattori, N. Nagano, “Numerical investigation of the stagnation problems of an iterative phase-retrieval algorithm,” Jpn. J. Opt. 21, 119–127 (1992).

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]

Other

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, 1987), Chap. 8, pp. 277–320.

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Figures (15)

Fig. 1
Fig. 1

Modified IOS in the case of 2×2 objects with L-shaped support.

Fig. 2
Fig. 2

Behavior of the ER in the case in which the original object f=(1, 1, 1), noise-free case. Open circles, positions of the initial objects; filled circles, positions of the reconstructed objects when g0(I) is used as the initial object; squares, positions of the convergence objects f, gcv(II), gcv(III), and gcv(IV).

Fig. 3
Fig. 3

Behavior of the ER in the case in which the original object f=(1, 2.5, 1.4); noise-free case.

Fig. 4
Fig. 4

Behavior of the ER in the case in which the original object f=(1, 3.5, 0.4); noise-free case.

Fig. 5
Fig. 5

Investigation of the mechanism of the assimilation of a territory into an adjacent one.

Fig. 6
Fig. 6

Rms error EFm versus the number of iterations m in the case in which the original object f=(1, 3.5, 0.4); g0(I) and g0(II) shown in Fig. 4, are used as the initial objects.

Fig. 7
Fig. 7

Behavior of the ER in the presence of noise; original object f=(1, 1, 1). (a) NR=9.8%, (b) NR=15.2%.

Fig. 8
Fig. 8

Rms error EFm versus the number of iterations m in the case in which the original object f=(1, 1, 1) and NR=15.2%; g0(IV), shown in Fig. 7(b), is used as the initial object.

Fig. 9
Fig. 9

Behavior of the ER in the case in which the original object f=(1, 3.5, 0.4) and NR=3.5%.

Fig. 10
Fig. 10

Distribution of zero objects of the first and the second kinds in the modified IOS, (b˜m, c˜m, d˜m), in the case of 3×2 objects with L-shaped support. In sections (a) b˜m=1, (b) b˜m =2, (c) b˜m=3.

Fig. 11
Fig. 11

Territories of the convergence objects in the case in which the original object f=(1, 1, 5.5, 0.8); noise-free case. In sections (a) b˜m=1, (b) b˜m=2, (c) b˜m=3.

Fig. 12
Fig. 12

Territories of the convergence objects in the case in which the original object f=(1, 1, 1, 1.3); noise-free case. In sections (a) b˜m=1, (b) b˜m=2.

Fig. 13
Fig. 13

Territories of the convergence objects in the case in which the original object f=(1, 1, 5.5, 0.8) and NR=7.8%. In sections (a) b˜m=1, (b) b˜m=2.

Fig. 14
Fig. 14

Investigation of the mechanism of the assimilation of territory II into territory III in the case in which the original object f=(1, 1, 5.5, 0.8). Territories in section (a) b˜m=1.147 in the noise-free case, (b) b˜m=1.419 in the case of NR=7.5%. The black dots indicated by arrows represent the position of the convergence object in territory II.

Fig. 15
Fig. 15

Territories of the convergence objects in the case in which the original object f=(1, 1, 1, 1.3) and NR=22%. In sections (a) b˜m=1, (b) b˜m=2.

Equations (42)

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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
(Fm2)gm(x, y)=2[gm(x, y)-gm(x, y)].
v(x, y)=-12(Fm2)gm(x, y).
gm(x, y)=gm(x, y)+v(x, y)
gm+1(x, y)=gm(x, y)+v(x, y)(x, y)D0(x, y)D .
gm+1(x, y)=gm(x, y)+vD(x, y).
gm(x, y)=gm(x, y).
v(x, y)=(MN)-1uv[|F(u, v)|-|Gm(u, v)|]exp[iθm(u, v)]×exp[i2π(ux/M+vy/N)].
v(x, y)=(MN)-1(u, v)(u0, v0),(-u0, -v0)[|F(u, v)|-|Gm(u, v)|]exp[iθm(u, v)]×exp[i2π(ux/Mx+vy/N)]+2(MN)-1|F(u0, v0)|cos[θm(u0, v0)+2π(u0x/M+v0y/N)]-2(MN)-1|Gm(u0, v0)|cos[θm(u0, v0)+2π(u0x/M+v0y/N)],
am0,bm0,cm0.
am-bm+cm=0,
am+bm-cm=0,
am-bm-cm=0,
am=bm,cm=0,
am=cm,bm=0,
bm=cm,am=0.
b˜m=bm/am,c˜m=cm/am,
a=1,
b=2.5+t,
c=1.4-t(0t1).
z=3-1/2|ac-bc+cc|.
EFm=Fm2(MN)-1uv|F(u, v)|21/2,
NR=uv[|Fn(u, v)|-|F(u, v)|]2uv|F(u, v)|21/2×100%,
am+bm+cm-dm=0,
am-bm+cm-dm=0,
am-bm+cm+dm=0,
am+bm2-cm2=0,32bm+32cm-dm=0,
am-bm2-cm2+dm=0,bm-cm=0,
am-bm2-cm2-dm=0,bm-cm=0,
am-bm2-cm2=0,32bm-32cm+dm=0,
am-bm2-cm2=0,32bm-32cm-dm=0.
b˜m=bm/am,c˜m=cm/am,d˜m=dm/am,
-12(F2)(+)g(x, y)=-g(+)(x, y)+g(+)(x, y),
-12(F2)(-)g(x, y)=-g(-)(x, y)+g(-)(x, y),
g(+)(x, y)=(MN)-1uv|F(u, v)|exp[iθ(u, v)]×exp[i2π(ux/M+vy/N)]=g¯(x, y)+(MN)-1|F(u0, v0)|exp[i0]×exp[i2π(u0x/M+v0y/N)]=g¯(x, y)+(MN)-1|F(u0, v0)|×exp[i2π(u0x/M+v0y/N)],
g(-)(x, y)=g¯(x, y)+(MN)-1|F(u0, v0)|exp[iπ]×exp[i2π(u0x/M+v0y/N)]=g¯(x, y)-(MN)-1|F(u0, v0)|×exp[i2π(u0x/M+v0y/N)],
g¯(x, y)=(MN)-1(u, v)(u0, v0)|F(u, v)|×exp[iθ(u, v)]exp[i2π(ux/M+vy/N)].
-12(F2)(+)g(x, y)=g¯(x, y)-g(+)(x, y)+(MN)-1|F(u0, v0)|×exp[i2π(u0x/M+v0y/N)],
-12(F2)(-)g(x, y)=g¯(x, y)-g(-)(x, y)-(MN)-1|F(u0, v0)|×exp[i2π(u0x/M+v0y/N)].

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