Abstract

A procedure is presented for designing distortion-invariant correlation filters. Optical correlation filters designed using this technique retain full position invariance. The filter design begins by finding the distortion-invariant modes (eigenfunctions) for a particular image. The input image, filter, and correlation response are all spectrally expanded in terms of these orthogonal eigenfunctions. An iterative technique between the spatial domain and the spectral domain is used to rephase the invariant modes so that a filter composed of a linear combination of modes has the proper overall invariance. The iterative technique also controls the information content of the filter by maintaining specified amplitudes for each invariant mode in the filter. Targets are detected by spanning the filter to determine points of constant amplitude.

© 1987 Optical Society of America

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References

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  1. R. W. Gerchberg, W. O. Saxton, Optik 34, 275 (1971).
  2. P. M. Hirsch, J. A. Jordan, L. B. Lesem, Patent U.S. No. 3,619,022 (November9, 1971).
  3. R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).
  4. N. C. Gallagher, B. Liu, Appl. Opt. 12, 2328 (1973).
    [CrossRef] [PubMed]
  5. R. W. Gerchberg, Opt. Acta 21, 709 (1974).
    [CrossRef]
  6. A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
    [CrossRef]
  7. D. C. Youla, H. Webb, IEEE Trans. Med. Imag. MIl, 81 (1982).
    [CrossRef]
  8. A. Vander Lugt, IRE Trans. Inf. Theory IT-10, 139 (1964).
    [CrossRef]
  9. G. F. Schils, D. W. Sweeney, J. Opt. Soc. Am. A 3, 1433 (1986).
    [CrossRef]
  10. Y.-N. Hsu, H. H. Arsenault, G. April, Appl. Opt. 21, 4012 (1982).
    [CrossRef] [PubMed]

1986

1982

1975

A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[CrossRef]

1974

R. W. Gerchberg, Opt. Acta 21, 709 (1974).
[CrossRef]

1973

1972

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

1971

R. W. Gerchberg, W. O. Saxton, Optik 34, 275 (1971).

1964

A. Vander Lugt, IRE Trans. Inf. Theory IT-10, 139 (1964).
[CrossRef]

April, G.

Arsenault, H. H.

Gallagher, N. C.

Gerchberg, R. W.

R. W. Gerchberg, Opt. Acta 21, 709 (1974).
[CrossRef]

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

R. W. Gerchberg, W. O. Saxton, Optik 34, 275 (1971).

Hirsch, P. M.

P. M. Hirsch, J. A. Jordan, L. B. Lesem, Patent U.S. No. 3,619,022 (November9, 1971).

Hsu, Y.-N.

Jordan, J. A.

P. M. Hirsch, J. A. Jordan, L. B. Lesem, Patent U.S. No. 3,619,022 (November9, 1971).

Lesem, L. B.

P. M. Hirsch, J. A. Jordan, L. B. Lesem, Patent U.S. No. 3,619,022 (November9, 1971).

Liu, B.

Papoulis, A.

A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[CrossRef]

Saxton, W. O.

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

R. W. Gerchberg, W. O. Saxton, Optik 34, 275 (1971).

Schils, G. F.

Sweeney, D. W.

Vander Lugt, A.

A. Vander Lugt, IRE Trans. Inf. Theory IT-10, 139 (1964).
[CrossRef]

Webb, H.

D. C. Youla, H. Webb, IEEE Trans. Med. Imag. MIl, 81 (1982).
[CrossRef]

Youla, D. C.

D. C. Youla, H. Webb, IEEE Trans. Med. Imag. MIl, 81 (1982).
[CrossRef]

Appl. Opt.

IEEE Trans. Circuits Syst.

A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[CrossRef]

IEEE Trans. Med. Imag.

D. C. Youla, H. Webb, IEEE Trans. Med. Imag. MIl, 81 (1982).
[CrossRef]

IRE Trans. Inf. Theory

A. Vander Lugt, IRE Trans. Inf. Theory IT-10, 139 (1964).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Acta

R. W. Gerchberg, Opt. Acta 21, 709 (1974).
[CrossRef]

Optik

R. W. Gerchberg, W. O. Saxton, Optik 34, 275 (1971).

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

Other

P. M. Hirsch, J. A. Jordan, L. B. Lesem, Patent U.S. No. 3,619,022 (November9, 1971).

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Figures (3)

Fig. 1
Fig. 1

Outline of the iterative synthesis technique.

Fig. 2
Fig. 2

Input scene for an optical correlator. A filter is designed to recognize the target at the lower right scaled continuously over the range 100 to 120%. The image at the top right is scaled by 120% of the image at the bottom right.

Fig. 3
Fig. 3

Digitally calculated ratio of the mean intensity to its standard deviation (〈I〉/σI) for each pixel of the correlation plane.

Equations (16)

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f ( x ) f [ A ( p ) x ] .
C f g ( p ) = R 2 f [ A ( p ) x ] g * ( x ) d x .
C f f ( p , q ) = R 2 f [ A ( p ) x ] f * [ A ( q ) x ] d x .
g ( x ) = P f [ A ( q ) x ] a * ( q ) d q ,
C f g ( p ) = P C f f ( p , q ) a ( q ) d q .
λ m ϕ m ( p ) = P C f f ( p , q ) ϕ m ( q ) d q .
C f g ( p ) = m c m ϕ m ( p ) ,
a ( q ) = m a m ϕ m ( q ) ,
f [ A ( p ) x ] = m f m ( x ) ϕ m ( p ) ,
f m ( x ) = P f [ A ( p ) x ] ϕ m * ( p ) d p .
g ( x ) = m a m * f m ( x ) ,
a m = c m / λ m ( λ m 0 ) .
| a m | 1 .
| C f g ( p ) | = b for p P .
C h g ( p ) = R 2 h [ A ( p ) x ] g * ( x ) d x p S , = R 2 f [ A ( q ) A ( p ) x ] g * ( x ) d x , q Q .
A ( q ) A ( p ) = A ( p ) .

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