Abstract

The iterative algorithm, projections onto constraint sets, is employed to generate spatial filters for pattern-recognition correlators. Based on all the training sets, all the filters are trained simultaneously.

© 1993 Optical Society of America

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References

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  1. J. Shamir, J. Rosen, U. Mahlab, J. H. Caulfield, Proc. Soc. Photo-Opt. Instrum. Eng. 40, 2 (1992).
  2. J. Rosen, J. Shamir, Opt. Lett. 16, 752 (1991).
    [CrossRef] [PubMed]
  3. C. F. Hester, D. P. Casasent, Appl. Opt. 19, 1758 (1980).
    [CrossRef] [PubMed]
  4. H. Stark, ed., Image Recovery Theory and Application, 1st ed. (Academic, New York, 1987), pp. 29 and 277.
  5. J. R. Fienup, Appl. Opt. 21, 2758 (1982).
    [CrossRef] [PubMed]
  6. J. Rosen, T. Kotzer, J. Shamir, Opt. Commun. 83, 10 (1991).
    [CrossRef]
  7. A. V. Oppenheim, J. S. Lim, Proc. IEEE 69, 529 (1981).
    [CrossRef]

1992

J. Shamir, J. Rosen, U. Mahlab, J. H. Caulfield, Proc. Soc. Photo-Opt. Instrum. Eng. 40, 2 (1992).

1991

J. Rosen, T. Kotzer, J. Shamir, Opt. Commun. 83, 10 (1991).
[CrossRef]

J. Rosen, J. Shamir, Opt. Lett. 16, 752 (1991).
[CrossRef] [PubMed]

1982

1981

A. V. Oppenheim, J. S. Lim, Proc. IEEE 69, 529 (1981).
[CrossRef]

1980

Casasent, D. P.

Caulfield, J. H.

J. Shamir, J. Rosen, U. Mahlab, J. H. Caulfield, Proc. Soc. Photo-Opt. Instrum. Eng. 40, 2 (1992).

Fienup, J. R.

Hester, C. F.

Kotzer, T.

J. Rosen, T. Kotzer, J. Shamir, Opt. Commun. 83, 10 (1991).
[CrossRef]

Lim, J. S.

A. V. Oppenheim, J. S. Lim, Proc. IEEE 69, 529 (1981).
[CrossRef]

Mahlab, U.

J. Shamir, J. Rosen, U. Mahlab, J. H. Caulfield, Proc. Soc. Photo-Opt. Instrum. Eng. 40, 2 (1992).

Oppenheim, A. V.

A. V. Oppenheim, J. S. Lim, Proc. IEEE 69, 529 (1981).
[CrossRef]

Rosen, J.

J. Shamir, J. Rosen, U. Mahlab, J. H. Caulfield, Proc. Soc. Photo-Opt. Instrum. Eng. 40, 2 (1992).

J. Rosen, T. Kotzer, J. Shamir, Opt. Commun. 83, 10 (1991).
[CrossRef]

J. Rosen, J. Shamir, Opt. Lett. 16, 752 (1991).
[CrossRef] [PubMed]

Shamir, J.

J. Shamir, J. Rosen, U. Mahlab, J. H. Caulfield, Proc. Soc. Photo-Opt. Instrum. Eng. 40, 2 (1992).

J. Rosen, J. Shamir, Opt. Lett. 16, 752 (1991).
[CrossRef] [PubMed]

J. Rosen, T. Kotzer, J. Shamir, Opt. Commun. 83, 10 (1991).
[CrossRef]

Appl. Opt.

Opt. Commun.

J. Rosen, T. Kotzer, J. Shamir, Opt. Commun. 83, 10 (1991).
[CrossRef]

Opt. Lett.

Proc. IEEE

A. V. Oppenheim, J. S. Lim, Proc. IEEE 69, 529 (1981).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

J. Shamir, J. Rosen, U. Mahlab, J. H. Caulfield, Proc. Soc. Photo-Opt. Instrum. Eng. 40, 2 (1992).

Other

H. Stark, ed., Image Recovery Theory and Application, 1st ed. (Academic, New York, 1987), pp. 29 and 277.

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Figures (4)

Fig. 1
Fig. 1

Block diagram of the POCS process.

Fig. 2
Fig. 2

Input training set.

Fig. 3
Fig. 3

(a) SDF plane, including two separated SDF’s (the real part). (b) The correlation plane after 100 iterations (the absolute value).

Fig. 4
Fig. 4

DR versus the number of iterations for the PEC learning system (solid curve) and for the linear correlator (dashed curve).

Equations (11)

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c ( x ) = - 1 { exp [ - j Φ ( u ) ] H ( u ) } ,
h ( x ) = - 1 { exp [ j Φ ( u ) ] { c ( x ) } } .
c ( x ) 2 d x = exp [ - j Φ ( u ) ] H ( u ) 2 d u = H ( u ) 2 d u = h ( x ) 2 d x .
P 1 [ c ( x ) ] = { T 1 exp [ j θ ( x ) ] if x R 1 and c ( x ) < T 1 T 2 exp [ j θ ( x ) ] if x R 2 and c ( x ) > T 2 c ( x ) otherwise ,
P 2 [ h ( x ) ] = { h ( x ) if x S 0 otherwise ,
e 1 , i c i ( x ) - P 1 [ c i ( x ) ] 2 d x , e 2 , i h i ( x ) - P 2 [ h i ( x ) ] 2 d x .
e 1 , i = P 2 [ h i ( x ) ] - h i + 1 ( x ) 2 d x .
e 1 , i h i + 1 ( x ) - P 2 [ h i + 1 ( x ) ] 2 d x = e 2 , i + 1 .
e 2 , i + 1 = h i + 1 ( x ) - P 2 [ h i + 1 ( x ) ] 2 d x = P 1 [ c i ( x ) ] - c i + 1 ( x ) 2 d x .
e 2 , i + 1 c i + 1 ( x ) - P 1 [ c i + 1 ( x ) ] 2 d x = e 1 , i + 1 .
γ a ( MIN x R 1 { c ( x ) } ) 2 / ( MAX x R 2 { c ( x ) } ) 2 .

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