Abstract

A superresolution method is presented with a conjugate-gradient procedure under a nonnegativity constraint. The method is based on the property of a nonnegative object that its band-limited spectrum is the autoconvolution of a spectrum with a half-band limit, and the method is capable of doubling the band limit of an image. Computer simulations are conducted to investigate the performance of the present method.

© 1996 Optical Society of America

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References

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1995

1994

O. L. Matson, IEEE Trans. Acoust. Speech Signal Process. 42, 156 (1994).

1993

1992

1989

1983

A. J. Levy, IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1337 (1983).
[CrossRef]

1974

R. W. Gerchberg, Opt. Acta 9, 709 (1974).
[CrossRef]

1964

Conan, J.-M.

Flannery, B. P.

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

Fried, D. L.

Gerchberg, R. W.

R. W. Gerchberg, Opt. Acta 9, 709 (1974).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

Harris, J. L.

Holmes, T. J.

Hunt, B. R.

Lane, R. G.

Levy, A. J.

A. J. Levy, IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1337 (1983).
[CrossRef]

Matson, O. L.

O. L. Matson, IEEE Trans. Acoust. Speech Signal Process. 42, 156 (1994).

Nadar, M. S.

Press, W. H.

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

Sementilli, P. J.

Teukolsky, S. A.

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

Thiebaut, E.

Vetterling, W. T.

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

IEEE Trans. Acoust. Speech Signal Process.

A. J. Levy, IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1337 (1983).
[CrossRef]

O. L. Matson, IEEE Trans. Acoust. Speech Signal Process. 42, 156 (1994).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

R. W. Gerchberg, Opt. Acta 9, 709 (1974).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

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 simulations of a compact four-point object: a, b, band-limited images with cutoff frequencies of 32 and 16 pixels, respectively; c, superresolved image; d, e, f, log-intensity plots of the power spectra of a, b, c, respectively, along the u axis. Images a–c are displayed with only the central 32 × 32 pixel region.

Fig. 2
Fig. 2

Computer simulation for a widely extended object: a, b, band-limited images with cutoff frequencies of 32 and 16 pixels, respectively; c, d, superresolved images with C2 = 32 and 28 pixels, respectively; e–h, log-intensity plots of power spectra of a–d along the u axis. In a–d black and white denote the maximum and the minimum, respectively.

Fig. 3
Fig. 3

Error metric and MSE versus iterations a, Fig. 1c, and b, for Fig. 2d.

Equations (14)

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o 1 ( x , y ) = f 1 2 ( x , y ) ,
O 1 ( u , υ ) = Ω F 1 ( u ' , υ ' ) F 1 ( u u ' , υ υ ' ) d u ' d υ ' ,
Ω = { ( u ' , υ ' ) | ( u ' , υ ' ) γ ( u u ' , υ υ ' ) γ } ,
O 1 ( u , υ ) = O 2 ( u , υ ) M 1 ( u , υ ) / M 2 ( u , υ ) .
O ˜ 1 ( u , υ ) = M 1 ( u , υ ) M 2 ( u , υ ) Ω ' F 2 ( u ' , υ ' ) × F 2 ( u u ' , υ υ ' ) d u ' d υ ' ,
O ˜ 1 ( u , υ ) = M 1 ( u , υ ) M 2 ( u , υ ) Ω ' F 2 ( u ' , υ ' ) × F 2 ( u u ' , υ υ ' ) d u ' d υ ' + N ( u , υ ) ,
E = Λ | O ˜ 1 ( u , υ ) M 1 ( u , υ ) M 2 ( u , υ ) Ω ' F 2 ( u ' , υ ' ) × F 2 ( u u ' , υ υ ' ) d u ' d υ ' | 2 d u d υ ,
E = Λ | O 1 ( u , υ ) M 1 ( u , υ ) M 2 ( u , υ ) × [ S ( u , υ ) + j T ( u , υ ) ] | 2 d u d υ ,
S ( u , υ ) = Ω ' A ( u ' , υ ' ) A ( u u ' , υ υ ' ) d u ' d υ ' Ω ' B ( u ' , υ ' ) B ( u u ' , υ υ ' ) d u ' d υ ' ,
T ( u , υ ) = Ω ' A ( u ' , υ ' ) B ( u u ' , υ υ ' ) d u ' d υ ' Ω ' B ( u ' , υ ' ) A ( u u ' , υ υ ' ) d u ' d υ ' .
E A ( u , υ ) = 4 Ω ' M 1 ( u ' , υ ' ) M 2 ( u ' , υ ' ) [ P ( u ' , υ ' ) A ( u ' u , υ ' υ ) + Q ( u ' , υ ' ) B ( u ' u , υ ' υ ) ] d u ' d υ ' ,
E B ( u , υ ) = 4 Ω ' M 1 ( u ' , υ ' ) M 2 ( u ' , υ ' ) [ Q ( u ' , υ ' ) A ( u ' u , υ ' υ ) P ( u ' , υ ' ) B ( u ' u , υ ' υ ) ] d u ' d υ ' ,
P ( u , υ ) Re [ O ˜ 1 ( u , υ ) ] S ( u , υ ) M 1 ( u , υ ) / M 2 ( u , υ ) ,
Q ( u , υ ) Im [ O ˜ 1 ( u , υ ) ] T ( u , υ ) M 1 ( u , υ ) / M 2 ( u , υ ) ,

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