Abstract

A general approach to the implementation of highly selective spatial filters for pattern recognition leads to a nonlinear optimization problem. Three optimization algorithms, hill climbing (direct search), simulated annealing, and the genetic algorithm, were investigated for implementation on hybrid electro-optical systems. Experimental results indicate the substantial superiority of the genetic algorithm in terms of operating speed and performance quality.

© 1992 Optical Society of America

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References

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  1. U. Mahlab, J. Rosen, J. Shamir, “Iterative generation of complex reference discriminant function in joint transform correlator,” Opt. Lett. 16, 330–332 (1991).
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    [Crossref]
  5. H. Rajbenbach, Y. Fainman, S. H. Lee “Optical implementation of an iterative algorithm for matrix inversion,” Appl. Opt. 26, 1024–1031 (1987).
    [Crossref] [PubMed]
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  7. J. H. Caulfield, J. H. Gruniger, J. E. Ludman, K. Steiglitz, H. Rabitz, J. Gelfand, E. Tsoni, “Bimodal optical computer,” Appl. Opt. 25, 3128–3131 (1986).
    [Crossref] [PubMed]
  8. J. Rosen, U. Mahlab, J. Shamir, “Adaptive learning with joint transform correlators,” Opt. Eng. 29, 1101–1106 (1990).
    [Crossref]
  9. M. Fleisher, U. Mahlab, J. Shamir “Entropy optimized filter for pattern recognition,” Appl. Opt. 29, 2091–2098 (1990).
    [Crossref] [PubMed]
  10. U. Mahlab, J. Shamir, “Optical pattern recognition based on a convex function”, J. Opt. Soc. Am. A 8, 1233–1239 (1991).
    [Crossref]
  11. B. V. K. Vijaya Kumar, Z. Bahri “Efficient algorithm for designing a ternary valued filter yielding maximum signal to noise ratio,” Appl. Opt. 28, 1919–1925 (1989).
    [Crossref]
  12. G. F. Schils, D. W. Sweeney, “Iterative technique for the synthesis of distortion-invariant optical correlation filters,” Opt. Lett. 12, 307–309 (1987).
    [Crossref] [PubMed]
  13. R. R. Kallman, “Construction of low noise optical correlation filters,” Appl. Opt. 25, 1032–1033 (1986).
    [Crossref] [PubMed]
  14. D. A. Jared, D. J. Ennis, “Inclusion of filter modulation in synthetic-discrimination function construction,” Appl. Opt. 28, 232–239 (1989).
    [Crossref] [PubMed]
  15. H. Bartelt, J. Horner, “Improving binary phase correlation filters using iterative techniques,” Appl. Opt. 24, 2894–2897 (1985).
    [Crossref] [PubMed]
  16. A. W. Robert, D. E. Veberg, Convex Functions (Academic, New York, 1973).
  17. A. Mahalanobis, B. V. K. V. Kumar, D. Casasent, “Minimum average correlation energy filter,” Appl. Opt. 26, 3633–3640 (1987).
    [Crossref] [PubMed]
  18. M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
    [Crossref] [PubMed]
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  20. U. Mahlab, J. Shamir, “Phase-only entropy-optimized filter by simulated annealing,” Opt. Lett. 14, 1168–1170 (1989).
    [Crossref] [PubMed]
  21. D. Lawrence, Genetic Algorithm and Simulated Annealing (Morgan Kaufmann, Los Altos, Calif., 1987).
  22. D. E. Goldberg, Genetic Algorithm in Search Optimization and Machine Learning (Addison-Wesley, Reading, Mass., 1989).
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  24. The CUE-2 is an image processing system based on a personal computer manufactured by GALAI Laboratories, Industrial Zone, Migdal Haemek, Israel.
  25. U. Mahlab, J. Rosen, J. Shamir, “Iterative generation of holograms on spatial light modulators,” Opt. Lett. 15, 556–558 (1990).
    [Crossref] [PubMed]
  26. N. Douklias, J. Shamir, “Relation between object position and auto-correlation spots in the VanderLugt filtering process. 2: Influence of the volume nature of the photographic emulsion,” Appl. Opt. 12, 364–367 (1973).
    [Crossref] [PubMed]

1991 (3)

1990 (3)

1989 (3)

1987 (4)

1986 (2)

1985 (1)

1980 (1)

1978 (1)

1973 (1)

Aarts, E. H. L.

P. J. M. van Luarhoven, E. H. L. Aarts, Simulated Annealing: Theory and Applications (Reidel, Dordrecht, The Netherlands, 1987).

Allebach, J. P.

Athale, R. A.

R. A. Athale, “Optical matrix processors,” in Optical and Hybrid Computing, H. S. Szu, ed., Proc. Soc. Photo-Opt. Instrum. Eng.634, 96–111 (1986).
[Crossref]

Avriel, M.

M. Avriel, Nonlinear Programming: Analysis and Methods (Prentice-Hall, Englewood Cliffs, N.J., 1976).

Bahri, Z.

Bartelt, H.

Casasent, D.

Caulfield, H. J.

Caulfield, J. H.

Cederquist, J.

Dias, A. R.

Douklias, N.

Ennis, D. J.

Fainman, Y.

Fleisher, M.

Gelfand, J.

Goldberg, D. E.

D. E. Goldberg, Genetic Algorithm in Search Optimization and Machine Learning (Addison-Wesley, Reading, Mass., 1989).

Goodman, J. W.

Gruniger, J. H.

Horner, J.

Jared, D. A.

Kallman, R. R.

Kumar, B. V. K. V.

Lawrence, D.

D. Lawrence, Genetic Algorithm and Simulated Annealing (Morgan Kaufmann, Los Altos, Calif., 1987).

Lee, S. H.

Ludman, J. E.

Mahalanobis, A.

Mahlab, U.

Rabitz, H.

Rajbenbach, H.

Robert, A. W.

A. W. Robert, D. E. Veberg, Convex Functions (Academic, New York, 1973).

Rosen, J.

Schils, G. F.

Seldowitz, M. A.

Shamir, J.

Steiglitz, K.

Sweeney, D. W.

Tsoni, E.

van Luarhoven, P. J. M.

P. J. M. van Luarhoven, E. H. L. Aarts, Simulated Annealing: Theory and Applications (Reidel, Dordrecht, The Netherlands, 1987).

Veberg, D. E.

A. W. Robert, D. E. Veberg, Convex Functions (Academic, New York, 1973).

Vijaya Kumar, B. V. K.

Woody, L. M.

Appl. Opt. (10)

H. Rajbenbach, Y. Fainman, S. H. Lee “Optical implementation of an iterative algorithm for matrix inversion,” Appl. Opt. 26, 1024–1031 (1987).
[Crossref] [PubMed]

J. H. Caulfield, J. H. Gruniger, J. E. Ludman, K. Steiglitz, H. Rabitz, J. Gelfand, E. Tsoni, “Bimodal optical computer,” Appl. Opt. 25, 3128–3131 (1986).
[Crossref] [PubMed]

M. Fleisher, U. Mahlab, J. Shamir “Entropy optimized filter for pattern recognition,” Appl. Opt. 29, 2091–2098 (1990).
[Crossref] [PubMed]

B. V. K. Vijaya Kumar, Z. Bahri “Efficient algorithm for designing a ternary valued filter yielding maximum signal to noise ratio,” Appl. Opt. 28, 1919–1925 (1989).
[Crossref]

R. R. Kallman, “Construction of low noise optical correlation filters,” Appl. Opt. 25, 1032–1033 (1986).
[Crossref] [PubMed]

D. A. Jared, D. J. Ennis, “Inclusion of filter modulation in synthetic-discrimination function construction,” Appl. Opt. 28, 232–239 (1989).
[Crossref] [PubMed]

H. Bartelt, J. Horner, “Improving binary phase correlation filters using iterative techniques,” Appl. Opt. 24, 2894–2897 (1985).
[Crossref] [PubMed]

A. Mahalanobis, B. V. K. V. Kumar, D. Casasent, “Minimum average correlation energy filter,” Appl. Opt. 26, 3633–3640 (1987).
[Crossref] [PubMed]

M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
[Crossref] [PubMed]

N. Douklias, J. Shamir, “Relation between object position and auto-correlation spots in the VanderLugt filtering process. 2: Influence of the volume nature of the photographic emulsion,” Appl. Opt. 12, 364–367 (1973).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

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

Opt. Eng. (1)

J. Rosen, U. Mahlab, J. Shamir, “Adaptive learning with joint transform correlators,” Opt. Eng. 29, 1101–1106 (1990).
[Crossref]

Opt. Lett. (6)

Other (7)

D. Lawrence, Genetic Algorithm and Simulated Annealing (Morgan Kaufmann, Los Altos, Calif., 1987).

D. E. Goldberg, Genetic Algorithm in Search Optimization and Machine Learning (Addison-Wesley, Reading, Mass., 1989).

P. J. M. van Luarhoven, E. H. L. Aarts, Simulated Annealing: Theory and Applications (Reidel, Dordrecht, The Netherlands, 1987).

The CUE-2 is an image processing system based on a personal computer manufactured by GALAI Laboratories, Industrial Zone, Migdal Haemek, Israel.

A. W. Robert, D. E. Veberg, Convex Functions (Academic, New York, 1973).

R. A. Athale, “Optical matrix processors,” in Optical and Hybrid Computing, H. S. Szu, ed., Proc. Soc. Photo-Opt. Instrum. Eng.634, 96–111 (1986).
[Crossref]

M. Avriel, Nonlinear Programming: Analysis and Methods (Prentice-Hall, Englewood Cliffs, N.J., 1976).

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Figures (10)

Fig. 1
Fig. 1

General architecture: OPs, optical signals; ELs, electric signals; O/E CONV, an optical-to-electrical converter.

Fig. 2
Fig. 2

Experimental system: 4-f optical correlator with a liquid crystal television at the Fourier plane and computer control. LP., input plane; F-P, Fourier transform plane; C-P, correlation plane; SLM, spatial light modulator.

Fig. 3
Fig. 3

Training set: the detected pattern T; the rejected pattern L.

Fig. 4
Fig. 4

Three-dimensional representation of the output correlation plane by using the direct search optimization procedure. The figures contain two correlation regions sampled by the CCD camera and separated by empty space. The left-hand side represents the correlation with the pattern to be detected, while the right-hand side corresponds to the rejected pattern. (a) Results after one iteration. (b)–(d) Intermediate results. (e) Final results.

Fig. 5
Fig. 5

Two-dimensional representation of the output correlation plane (including the zero order) achieved by the GA. The filter detected T and rejected L.

Fig. 6
Fig. 6

Training set for a time convergence comparison: E to be detected, P to be rejected.

Fig. 7
Fig. 7

Convergence profiles sharing the cost function versus iteration number. (Each iteration consists of a single filter update.) Solid curves are for the GA, while the dashed–dotted fines correspond to (a) the HC and (b) the SA algorithms.

Fig. 8
Fig. 8

Convergence profiles: (a) HC, (b) SA.

Fig. 9
Fig. 9

Gray level input patterns: (a) picture to be detected, (b) picture to be rejected.

Fig. 10
Fig. 10

(a) Cross section of the correlation peaks for the patterns in Fig. 9. (b) Cross section of correlation for the picture in Fig. 9(a) presented three times.

Tables (1)

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Table I Optimization Procedures

Equations (18)

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c ( x 0 , y 0 ) = f ( x , y ) h * ( x + x 0 , y + y 0 ) d x d y ,
Φ ( x , y ) = L [ c ( x , y ) ] L [ c ( x , y ) ] d x d y ,
S = Ψ [ Φ ( x , y ) ] d x d y ,
Φ D ( m , n ) = { 1 at ( m = k n = l ) ( domain of Φ ) 0 otherwise ,
Φ R ( m , n ) = 1 ( 2 N 1 ) 2 , ( m , n ) ,
Ψ ( x ) = x log x ,
S min D = 0 , S max R = log 1 ( 2 N 1 ) 2 ,
M = f n D S D f n R S R
M min = M [ h GEF ( i , j ) ] .
M ( t ) = M [ h ( t ) ( i , j ) ] ,
Δ M ( t + 1 ) = M [ h ( t + 1 ) ( i , j ) ] M [ h ( t ) ( i , j ) ] .
P r accept = exp [ Δ M ( t + 1 ) T ] ,
θ = 1 m i = 1 m M i .
θ θ + 1 m ( M c M d ) .
σ t = { 1 t j = 1 t [ Δ M ( j ) 1 t i = 1 t Δ M ( i ) ] 2 } 1 / 2 .
Δ exp ( t + 1 ) = Δ M ( t + 1 ) Δ M ( t ) ,
| Δ exp ( t + 1 ) | < σ t .
Δ exp ( t + 1 ) < 0 ,

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