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 (2)

1994 (1)

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

1993 (1)

1992 (1)

1989 (1)

1983 (1)

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

1974 (1)

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

1964 (1)

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. (2)

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. (1)

J. Opt. Soc. Am. A (5)

Opt. Acta (1)

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

Other (2)

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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