Abstract

It is shown that a band-limited spectrum of a nonnegative object can be represented as the autoconvolution of a spectrum whose cutoff frequency is half of the band limit. This permits spectral-magnitude-only reconstruction with a nonnegativity constraint without a support constraint. A magnitude-only reconstruction method using a conjugate-gradient procedure is developed and is investigated by computer simulations.

© 1996 Optical Society of America

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References

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  1. R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).
  2. J. R. Fienup, Opt. Lett. 3, 27 (1978).
    [CrossRef] [PubMed]
  3. J. R. Fienup, Appl. Opt. 21, 2758 (1982).
    [CrossRef] [PubMed]
  4. M. H. Hayes, IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140 (1982).
    [CrossRef]
  5. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).
  6. A. Labeyrie, Astron. Astrophys. 6, 85 (1970).
  7. B. L. McGlamery, Proc. SPIE 74, 225 (1976).
  8. J. H. Seldin, J. R. Fienup, J. Opt. Soc. Am. A 7, 428 (1990).
    [CrossRef]

1990 (1)

1982 (2)

J. R. Fienup, Appl. Opt. 21, 2758 (1982).
[CrossRef] [PubMed]

M. H. Hayes, IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140 (1982).
[CrossRef]

1978 (1)

1976 (1)

B. L. McGlamery, Proc. SPIE 74, 225 (1976).

1972 (1)

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

1970 (1)

A. Labeyrie, Astron. Astrophys. 6, 85 (1970).

Fienup, J. R.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

Gerchberg, R. W.

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

Hayes, M. H.

M. H. Hayes, IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140 (1982).
[CrossRef]

Labeyrie, A.

A. Labeyrie, Astron. Astrophys. 6, 85 (1970).

McGlamery, B. L.

B. L. McGlamery, Proc. SPIE 74, 225 (1976).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

Saxton, W. O.

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

Seldin, J. H.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

Appl. Opt. (1)

Astron. Astrophys. (1)

A. Labeyrie, Astron. Astrophys. 6, 85 (1970).

IEEE Trans. Acoust. Speech Signal Process (1)

M. H. Hayes, IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140 (1982).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Optik (1)

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

Proc. SPIE (1)

B. L. McGlamery, Proc. SPIE 74, 225 (1976).

Other (1)

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

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Figures (3)

Fig. 1
Fig. 1

Computer simulation for the noise-free case: (a) disklike object; (b) a diffraction-limited image of the object, (c) the power spectrum of (b) on a log-intensity scale, (c) the reconstructed image. Only the central 32 × 32 regions are displayed in images (a), (b), and (d). In halftone images, black is the maximum and white is the minimum.

Fig. 2
Fig. 2

Computer simulation with use of the noisy power spectrum: (a) the noisy power spectrum obtained from 100 specklegrams as shown in (b) on a log-intensity scale, (c) the image reconstructed by the present method, (d) the image reconstructed by the error-reduction algorithm. Only the central 32 × 32 regions are displayed in images (c) and (d).

Fig. 3
Fig. 3

(a) Error metrics and (b) MSE versus iterations in reconstructions of Figs. 1(d), 2(c), and 2(d), which corresponds to solid, dashed, and dotted curves, respectively.

Equations (12)

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o ( x , y ) = f 2 ( x , y ) ,
O ( u , ν ) = Ω F ( u , ν ) F ( u u , ν ν ) d u d ν ,
Ω = [ ( u , ν ) | ( u , ν ) γ ( u u , v v ) γ ] ,
| O ( u , ν ) | 2 = | Ω F ( u , ν ) F ( u u , ν ν ) d u d ν | 2 .
| O ˜ ( u , ν ) | 2 = | Ω F ( u , ν ) F ( u u , ν ν ) d u d ν | 2 + N ( u , ν ) .
E = Λ [ | O ˜ ( u , ν ) | 2 | Ω F ( u , ν ) F ( u u , v ν ) × d u d ν | 2 ] 2 d u d v ,
O ( u , ν ) = Ω A ( u , ν ) A ( u u , ν ν ) d u d ν Ω B ( u , ν ) B ( u u , ν ν ) d u d ν + j [ Ω A ( u , ν ) B ( u u , ν ν ) d u d ν + Ω B ( u , ν ) A ( u u , ν ν ) d u d ν ] . S ( u , ν ) + j T ( u , ν ) .
E = Λ [ | O ˜ ( u , ν ) | 2 S 2 ( u , ν ) T 2 ( u , ν ) ] 2 d u d ν .
δ E δ A ( u , ν ) = 8 Ω [ | O ˜ ( u , ν ) | 2 S 2 ( u , ν ) T 2 ( u , ν ) ] [ S ( u , ν ) A ( u u , ν ν ) + T ( u , ν ) B ( u u , ν ν ) ] d u d ν ,
δ E δ A ( u , v ) = δ E δ A ( u , ν ) ,
δ E δ B ( u , ν ) = 8 Ω [ | O ˜ ( u , ν ) | 2 S 2 ( u , ν ) T 2 ( u , ν ) ] [ S ( u , ν ) B ( u u , ν ν ) + T ( u , ν ) A ( u u , ν ν ) ] d u d ν ,
δ E δ B ( u , v ) = δ E δ B ( u , v ) .

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