Abstract

We present a digital method for solving the phase-retrieval problem of optical-coherence theory: the reconstruction of a general object from the modulus of its Fourier transform. This technique should be useful for obtaining high-resolution imagery from interferometer data.

© 1978 Optical Society of America

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References

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  1. R. H. T. Bates, IEEE Trans. Comput. C-24, 449 (1975).
    [CrossRef]
  2. A. Labeyrie, Astron. Astrophys. 6, 85 (1970); D. Y. Gezari, A. Labeyrie, R. V. Stachnik, Astrophys. Lett. 173, L1 (1972).
    [CrossRef]
  3. D. G. Currie, S. L. Knapp, K. M. Liewer, Astrophys. J. 187, 131 (1974).
    [CrossRef]
  4. For a list of references, see Ref. 4 in D. Kohler, L. Mandel, J. Opt. Soc. Am. 63, 134 (1973).
  5. B. R. Frieden, D. G. Currie, J. Opt. Soc. Am. 66, 1111A (1976).
    [CrossRef]
  6. R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).
  7. N. C. Gallagher, B. Liu, Appl. Opt. 12, 2328 (1973); R. W. Gerchberg, Opt. Acta 21, 709 (1974).
    [CrossRef] [PubMed]
  8. B. Liu, N. C. Gallagher, Appl. Opt. 13, 2470 (1974).
    [CrossRef] [PubMed]
  9. J. R. Fienup, J. Opt. Soc. Am. 64, 1395A (1974).
  10. J. R. Fienup, Ph.D. thesis, Stanford University, May1975 (University Microfilms No. 75-25523), Chap. 5.

1976

B. R. Frieden, D. G. Currie, J. Opt. Soc. Am. 66, 1111A (1976).
[CrossRef]

1975

R. H. T. Bates, IEEE Trans. Comput. C-24, 449 (1975).
[CrossRef]

1974

D. G. Currie, S. L. Knapp, K. M. Liewer, Astrophys. J. 187, 131 (1974).
[CrossRef]

B. Liu, N. C. Gallagher, Appl. Opt. 13, 2470 (1974).
[CrossRef] [PubMed]

J. R. Fienup, J. Opt. Soc. Am. 64, 1395A (1974).

1973

For a list of references, see Ref. 4 in D. Kohler, L. Mandel, J. Opt. Soc. Am. 63, 134 (1973).

N. C. Gallagher, B. Liu, Appl. Opt. 12, 2328 (1973); R. W. Gerchberg, Opt. Acta 21, 709 (1974).
[CrossRef] [PubMed]

1972

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

1970

A. Labeyrie, Astron. Astrophys. 6, 85 (1970); D. Y. Gezari, A. Labeyrie, R. V. Stachnik, Astrophys. Lett. 173, L1 (1972).
[CrossRef]

Bates, R. H. T.

R. H. T. Bates, IEEE Trans. Comput. C-24, 449 (1975).
[CrossRef]

Currie, D. G.

B. R. Frieden, D. G. Currie, J. Opt. Soc. Am. 66, 1111A (1976).
[CrossRef]

D. G. Currie, S. L. Knapp, K. M. Liewer, Astrophys. J. 187, 131 (1974).
[CrossRef]

Fienup, J. R.

J. R. Fienup, J. Opt. Soc. Am. 64, 1395A (1974).

J. R. Fienup, Ph.D. thesis, Stanford University, May1975 (University Microfilms No. 75-25523), Chap. 5.

Frieden, B. R.

B. R. Frieden, D. G. Currie, J. Opt. Soc. Am. 66, 1111A (1976).
[CrossRef]

Gallagher, N. C.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

Knapp, S. L.

D. G. Currie, S. L. Knapp, K. M. Liewer, Astrophys. J. 187, 131 (1974).
[CrossRef]

Kohler, D.

For a list of references, see Ref. 4 in D. Kohler, L. Mandel, J. Opt. Soc. Am. 63, 134 (1973).

Labeyrie, A.

A. Labeyrie, Astron. Astrophys. 6, 85 (1970); D. Y. Gezari, A. Labeyrie, R. V. Stachnik, Astrophys. Lett. 173, L1 (1972).
[CrossRef]

Liewer, K. M.

D. G. Currie, S. L. Knapp, K. M. Liewer, Astrophys. J. 187, 131 (1974).
[CrossRef]

Liu, B.

Mandel, L.

For a list of references, see Ref. 4 in D. Kohler, L. Mandel, J. Opt. Soc. Am. 63, 134 (1973).

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

Appl. Opt.

Astron. Astrophys.

A. Labeyrie, Astron. Astrophys. 6, 85 (1970); D. Y. Gezari, A. Labeyrie, R. V. Stachnik, Astrophys. Lett. 173, L1 (1972).
[CrossRef]

Astrophys. J.

D. G. Currie, S. L. Knapp, K. M. Liewer, Astrophys. J. 187, 131 (1974).
[CrossRef]

IEEE Trans. Comput.

R. H. T. Bates, IEEE Trans. Comput. C-24, 449 (1975).
[CrossRef]

J. Opt. Soc. Am.

For a list of references, see Ref. 4 in D. Kohler, L. Mandel, J. Opt. Soc. Am. 63, 134 (1973).

B. R. Frieden, D. G. Currie, J. Opt. Soc. Am. 66, 1111A (1976).
[CrossRef]

J. R. Fienup, J. Opt. Soc. Am. 64, 1395A (1974).

Optik

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

Other

J. R. Fienup, Ph.D. thesis, Stanford University, May1975 (University Microfilms No. 75-25523), Chap. 5.

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Figures (4)

Fig. 1
Fig. 1

(a) Block diagram of the error-reduction approach; (b) block diagram of the system for the input–output concept.

Fig. 2
Fig. 2

Reconstruction results: test object and 15 other solutions.

Fig. 3
Fig. 3

(a) Test object; (b) modulus of its Fourier transform; (c) initial estimate of the object (first test); (d)–(f) reconstruction results—number of iterations: (d) 20, (e) 230, (f) 600; (g) initial estimate of the object (second test); (h)–(i) reconstruction results—number of iterations: (h) 2, (i) 215.

Fig. 4
Fig. 4

(a) Reconstruction result for 2.2% rms noise; (b) reconstruction result for 11% rms noise.

Equations (5)

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F ( u ) = F ( u ) exp [ i ψ ( u ) ] = F { f ( x ) } - f ( x ) exp ( - i 2 π u · x ) d x ,
g k + 1 ( x ) = { g k ( x ) , x γ 0 , x γ ,
E F 2 = - [ G k ( u ) - F ( u ) ] 2 d u - F ( u ) 2 d u ,
E 0 2 = γ [ g k ( x ) ] 2 d x - [ g k ( x ) ] 2 d x .
g k + 1 ( x ) = { g k ( x ) , x γ g k ( x ) - β g k ( x ) , x γ ,

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