Abstract

A new approach is given for the problem of reconstruction of phase from modulus data. A set of Wiener-filter functions is formed that multiply, in turn, displaced versions of the modulus data (in frequency space) such that the sum is a minimum L2-error norm solution for the object. The modulus data are permitted to contain both noise and signal (object) components. The required statistics are power spectra of the signal and noise, and correlations between modulus data at given frequencies and complex object spectral values at adjacent frequencies. In a numerical simulation, a 3 × 3 filter array is used to reconstruct any member of an object class consisting of 16 pictures of space shuttles in various combinations. The 16 pictures are used as a learning set to form the required power spectra and correlations mentioned above. Reconstructions are formed in the presence of data noise, data gaps, and filter-construction noise, in varying amounts. Results are encouraging in that the space shuttle images are always recognizable.

© 1992 Optical Society of America

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References

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  1. D. G. Currie, “On a detection scheme for an amplitude interferometer,” in Synthetic-Aperture Optics, Volume 2, Woods Hole Summer Study, J. W. Goodman, ed. Defense Documentation Center, Alexandria, Va., 1967, App. II.
  2. J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” Image Recovery, H. Stark, ed. (Academic, Orlando, Fla., 1987), Chap. 7.
  3. H. A. Ferwerda, Inverse Source Problems, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Chap. 2.
  4. L. C. Liu, S. H. Risbud, “Analysis of TEM image contrast of quantum-dot semiconductor clusters in glasses,” Philos. Mag. Lett. 61, 327–332 (1990).
    [Crossref]
  5. J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
    [Crossref]
  6. B. R. Frieden, D. G. Currie, “On unfolding the autocorrelation function,” J. Opt. Soc. Am. 66, 1111A (1976).
  7. K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
    [Crossref]
  8. A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
    [Crossref] [PubMed]
  9. N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series (Wiley, New York, 1949).
  10. B. R. Frieden, Probability, Statistical Optics and Data Testing, 2nd ed. (Springer-Verlag, New York, 1991).
    [Crossref]
  11. R. A. Messner, H. H. Szu, “Simultaneous image processing and feature extraction for two-dimensional non-uniform sensors,” in Intelligent Robots: Third International Conference on Robot Vision and Sensory Controls, D. P. Casasent, E. L. Hall, eds., Proc. Soc. Photo-Opt. Instrum. Eng.449, 693–710 (1983).

1990 (1)

L. C. Liu, S. H. Risbud, “Analysis of TEM image contrast of quantum-dot semiconductor clusters in glasses,” Philos. Mag. Lett. 61, 327–332 (1990).
[Crossref]

1983 (1)

1981 (1)

J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
[Crossref]

1976 (1)

B. R. Frieden, D. G. Currie, “On unfolding the autocorrelation function,” J. Opt. Soc. Am. 66, 1111A (1976).

1974 (1)

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[Crossref]

Currie, D. G.

B. R. Frieden, D. G. Currie, “On unfolding the autocorrelation function,” J. Opt. Soc. Am. 66, 1111A (1976).

D. G. Currie, “On a detection scheme for an amplitude interferometer,” in Synthetic-Aperture Optics, Volume 2, Woods Hole Summer Study, J. W. Goodman, ed. Defense Documentation Center, Alexandria, Va., 1967, App. II.

Dainty, J. C.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” Image Recovery, H. Stark, ed. (Academic, Orlando, Fla., 1987), Chap. 7.

Ferwerda, H. A.

H. A. Ferwerda, Inverse Source Problems, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Chap. 2.

Fienup, J. R.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” Image Recovery, H. Stark, ed. (Academic, Orlando, Fla., 1987), Chap. 7.

Frieden, B. R.

B. R. Frieden, D. G. Currie, “On unfolding the autocorrelation function,” J. Opt. Soc. Am. 66, 1111A (1976).

B. R. Frieden, Probability, Statistical Optics and Data Testing, 2nd ed. (Springer-Verlag, New York, 1991).
[Crossref]

Knox, K. T.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[Crossref]

Liu, L. C.

L. C. Liu, S. H. Risbud, “Analysis of TEM image contrast of quantum-dot semiconductor clusters in glasses,” Philos. Mag. Lett. 61, 327–332 (1990).
[Crossref]

Lohmann, A. W.

Messner, R. A.

R. A. Messner, H. H. Szu, “Simultaneous image processing and feature extraction for two-dimensional non-uniform sensors,” in Intelligent Robots: Third International Conference on Robot Vision and Sensory Controls, D. P. Casasent, E. L. Hall, eds., Proc. Soc. Photo-Opt. Instrum. Eng.449, 693–710 (1983).

Risbud, S. H.

L. C. Liu, S. H. Risbud, “Analysis of TEM image contrast of quantum-dot semiconductor clusters in glasses,” Philos. Mag. Lett. 61, 327–332 (1990).
[Crossref]

Szu, H. H.

R. A. Messner, H. H. Szu, “Simultaneous image processing and feature extraction for two-dimensional non-uniform sensors,” in Intelligent Robots: Third International Conference on Robot Vision and Sensory Controls, D. P. Casasent, E. L. Hall, eds., Proc. Soc. Photo-Opt. Instrum. Eng.449, 693–710 (1983).

Thompson, B. J.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[Crossref]

Walker, J. G.

J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
[Crossref]

Weigelt, G.

Wiener, N.

N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series (Wiley, New York, 1949).

Wirnitzer, B.

Appl. Opt. (1)

Astrophys. J. Lett. (1)

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[Crossref]

J. Opt. Soc. Am. (1)

B. R. Frieden, D. G. Currie, “On unfolding the autocorrelation function,” J. Opt. Soc. Am. 66, 1111A (1976).

Opt. Acta (1)

J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
[Crossref]

Philos. Mag. Lett. (1)

L. C. Liu, S. H. Risbud, “Analysis of TEM image contrast of quantum-dot semiconductor clusters in glasses,” Philos. Mag. Lett. 61, 327–332 (1990).
[Crossref]

Other (6)

D. G. Currie, “On a detection scheme for an amplitude interferometer,” in Synthetic-Aperture Optics, Volume 2, Woods Hole Summer Study, J. W. Goodman, ed. Defense Documentation Center, Alexandria, Va., 1967, App. II.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” Image Recovery, H. Stark, ed. (Academic, Orlando, Fla., 1987), Chap. 7.

H. A. Ferwerda, Inverse Source Problems, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Chap. 2.

N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series (Wiley, New York, 1949).

B. R. Frieden, Probability, Statistical Optics and Data Testing, 2nd ed. (Springer-Verlag, New York, 1991).
[Crossref]

R. A. Messner, H. H. Szu, “Simultaneous image processing and feature extraction for two-dimensional non-uniform sensors,” in Intelligent Robots: Third International Conference on Robot Vision and Sensory Controls, D. P. Casasent, E. L. Hall, eds., Proc. Soc. Photo-Opt. Instrum. Eng.449, 693–710 (1983).

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Figures (8)

Fig. 1
Fig. 1

Object training set.

Fig. 2
Fig. 2

The nine modulus filters.

Fig. 3
Fig. 3

The nine phase filters.

Fig. 4
Fig. 4

Nine-filter outputs (no noise).

Fig. 5
Fig. 5

Nine-filter outputs (10% noise).

Fig. 6
Fig. 6

Nine-filter outputs (20% noise).

Fig. 7
Fig. 7

Outputs, with 10% noise in the filters.

Fig. 8
Fig. 8

Outputs, from data with 50% obscuration.

Equations (13)

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O ( ω ) = ( 2 π ) 1 / 2 d x o ( x ) exp ( j ω x ) , j = 1 .
D ( ω ) | O ( ω ) + N ( ω ) | , | ω | Ω ,
Y n ( ω ) , n = 1 , , N ,
O ˆ ( ω ) n = 1 N Y n ( ω ) D ( ω + m Δ ω ) , m n 1 ( N 1 ) / 2
O ˆ ( ω ) = Y 1 ( ω ) D ( ω Δ ω ) + Y 2 ( ω ) D ( ω ) + Y 3 ( ω ) D ( ω + Δ ω ) .
e 2 Ω Ω d ω | O ˆ ( ω ) O ( ω ) | 2 = minimum , Ω Ω d ω E .
Y n * ( ω ) | O ( ω + Δ ω ) + N ( ω + Δ ω ) | O ( ω ) .
1 M m = 1 M | O m ( ω + Δ ω ) + N m ( ω + Δ ω ) | O m ( ω ) .
E Y n * = 0 , n = 1 , , N .
e 2 = d ω | Y | O + N | O | 2 = d ω [ YY * | O + N | 2 + | O | 2 Y | O + N | O * Y * | O + N | O ] d ω E .
E Y * = Y | O + N | 2 | O + N | O 0
Y ( ω ) = | O ( ω ) + N ( ω ) | O ( ω ) | O + N | 2 .
Y ( ω ) = | O ( ω ) | O ( ω ) | O ( ω ) | 2 .

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