Abstract

Spatial filters based on the properties of convex functions can be generated directly on a hybrid electro-optical system. As an example, simulated annealing was used to design highly selective spatial filters. Laboratory experiments demonstrated efficient pattern recognition and class discrimination. Several procedures for designing synthetic discriminant functions discussed in the literature are shown to be special cases of the present procedure.

© 1991 Optical Society of America

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References

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  1. A. B. VanderLugt, “Signal detection by complex spatial filtering,”IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
  2. C. F. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
    [Crossref] [PubMed]
  3. D. Casasent, “Unified synthetic discriminant function computational formulation,” Appl. Opt. 23, 1620–1627 (1984).
    [Crossref] [PubMed]
  4. V. Sharma, D. Casasent, “Optimal linear discriminant functions,” in Solid State Imagers and Their Applications, G. J. Declerk, ed., Proc. Soc. Photo-Opt. Instrum. Eng.591, 50–55 (1985).
  5. Z. Bahri, B. V. K. Vijaya Kumar, “Generalized synthetic discriminant functions,” J. Opt. Soc. Am. A 5, 562–571 (1988).
    [Crossref]
  6. D. Casasent, W. T. Chang, “Correlation synthetic discrimination functions,” Appl. Opt. 25, 2343–2350 (1986).
    [Crossref] [PubMed]
  7. A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
    [Crossref] [PubMed]
  8. B. V. K. Vijaya Kumar, “Minimum-variance synthetic discrimination functions,” J. Opt. Soc. Am. A 3, 1579–1584 (1986).
    [Crossref]
  9. R. Kallman, “Construction of low noise optical correlation filters,” Appl. Opt. 25, 1032–1033 (1986).
    [Crossref] [PubMed]
  10. M. Fleisher, U. Mahlab, J. Shamir, “Entropy optimized filter for pattern recognition,” Appl. Opt. 29, 2091–2098 (1990).
    [Crossref] [PubMed]
  11. U. Mahlab, J. Shamir, “Phase-only entropy-optimized filter generated by simulated annealing,” Opt. Lett. 14, 1168–1170 (1989).
    [Crossref] [PubMed]
  12. A. W. Robert, D. E. Veberg, Convex Function (Academic, New York, 1973).
  13. B. V. K. Vijaya Kumar, Z. Bahri, L. Hassebrook, “Review of SDF algorithm,” in Real-Time Signal Processing for Industrial Applications, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.960, 18–29 (1988).
  14. C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (University of Illinois, Urbana, Ill., 1949).
  15. J. Ziv, M. Zakai, “On functionals satisfying a data-processing theorem,”IEEE Trans. Inf. Theory IT-19, 275–283 (1973).
    [Crossref]
  16. J. Rosen, U. Mahlab, J. Shamir, “Adaptive learning with joint transform correlator,” Opt. Eng. 29, 1101–1106 (1990).
    [Crossref]
  17. P. J. M. Van Luahoven, E. H. L. Aarts, Simulated Annealing: Theory and Applications (Reidel, Dordrecht, The Netherlands, 1987).
  18. S. Sola, E. Levin, M. Fleisher, “Accelerated learning in layered neural networks,” Complex Syst. 2, 625–640 (1988).
  19. G. Zalman, J. Shamir, “Maximum discrimination filter,” J. Opt. Soc. Am. A 8, 814–821 (1991).
    [Crossref]
  20. U. Mahlab, J. Rosen, J. Shamir, “Iterative generation of complex RDF in joint transform correlators,” Opt. Lett. (to be published.)
  21. U. Mahlab, J. Shamir, J. H. Caulfield, “Genetic algorithm for optical pattern recognition,” Opt. Lett. (to be published).
    [PubMed]

1991 (1)

1990 (2)

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

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

1989 (1)

1988 (2)

Z. Bahri, B. V. K. Vijaya Kumar, “Generalized synthetic discriminant functions,” J. Opt. Soc. Am. A 5, 562–571 (1988).
[Crossref]

S. Sola, E. Levin, M. Fleisher, “Accelerated learning in layered neural networks,” Complex Syst. 2, 625–640 (1988).

1987 (1)

1986 (3)

1984 (1)

1980 (1)

1973 (1)

J. Ziv, M. Zakai, “On functionals satisfying a data-processing theorem,”IEEE Trans. Inf. Theory IT-19, 275–283 (1973).
[Crossref]

1964 (1)

A. B. VanderLugt, “Signal detection by complex spatial filtering,”IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

Aarts, E. H. L.

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

Bahri, Z.

Z. Bahri, B. V. K. Vijaya Kumar, “Generalized synthetic discriminant functions,” J. Opt. Soc. Am. A 5, 562–571 (1988).
[Crossref]

B. V. K. Vijaya Kumar, Z. Bahri, L. Hassebrook, “Review of SDF algorithm,” in Real-Time Signal Processing for Industrial Applications, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.960, 18–29 (1988).

Casasent, D.

Caulfield, J. H.

U. Mahlab, J. Shamir, J. H. Caulfield, “Genetic algorithm for optical pattern recognition,” Opt. Lett. (to be published).
[PubMed]

Chang, W. T.

Fleisher, M.

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

S. Sola, E. Levin, M. Fleisher, “Accelerated learning in layered neural networks,” Complex Syst. 2, 625–640 (1988).

Hassebrook, L.

B. V. K. Vijaya Kumar, Z. Bahri, L. Hassebrook, “Review of SDF algorithm,” in Real-Time Signal Processing for Industrial Applications, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.960, 18–29 (1988).

Hester, C. F.

Kallman, R.

Levin, E.

S. Sola, E. Levin, M. Fleisher, “Accelerated learning in layered neural networks,” Complex Syst. 2, 625–640 (1988).

Mahalanobis, A.

Mahlab, U.

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

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

U. Mahlab, J. Shamir, “Phase-only entropy-optimized filter generated by simulated annealing,” Opt. Lett. 14, 1168–1170 (1989).
[Crossref] [PubMed]

U. Mahlab, J. Rosen, J. Shamir, “Iterative generation of complex RDF in joint transform correlators,” Opt. Lett. (to be published.)

U. Mahlab, J. Shamir, J. H. Caulfield, “Genetic algorithm for optical pattern recognition,” Opt. Lett. (to be published).
[PubMed]

Robert, A. W.

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

Rosen, J.

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

U. Mahlab, J. Rosen, J. Shamir, “Iterative generation of complex RDF in joint transform correlators,” Opt. Lett. (to be published.)

Shamir, J.

G. Zalman, J. Shamir, “Maximum discrimination filter,” J. Opt. Soc. Am. A 8, 814–821 (1991).
[Crossref]

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

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

U. Mahlab, J. Shamir, “Phase-only entropy-optimized filter generated by simulated annealing,” Opt. Lett. 14, 1168–1170 (1989).
[Crossref] [PubMed]

U. Mahlab, J. Shamir, J. H. Caulfield, “Genetic algorithm for optical pattern recognition,” Opt. Lett. (to be published).
[PubMed]

U. Mahlab, J. Rosen, J. Shamir, “Iterative generation of complex RDF in joint transform correlators,” Opt. Lett. (to be published.)

Shannon, C. E.

C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (University of Illinois, Urbana, Ill., 1949).

Sharma, V.

V. Sharma, D. Casasent, “Optimal linear discriminant functions,” in Solid State Imagers and Their Applications, G. J. Declerk, ed., Proc. Soc. Photo-Opt. Instrum. Eng.591, 50–55 (1985).

Sola, S.

S. Sola, E. Levin, M. Fleisher, “Accelerated learning in layered neural networks,” Complex Syst. 2, 625–640 (1988).

Van Luahoven, P. J. M.

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

VanderLugt, A. B.

A. B. VanderLugt, “Signal detection by complex spatial filtering,”IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

Veberg, D. E.

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

Vijaya Kumar, B. V. K.

Weaver, W.

C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (University of Illinois, Urbana, Ill., 1949).

Zakai, M.

J. Ziv, M. Zakai, “On functionals satisfying a data-processing theorem,”IEEE Trans. Inf. Theory IT-19, 275–283 (1973).
[Crossref]

Zalman, G.

Ziv, J.

J. Ziv, M. Zakai, “On functionals satisfying a data-processing theorem,”IEEE Trans. Inf. Theory IT-19, 275–283 (1973).
[Crossref]

Appl. Opt. (6)

Complex Syst. (1)

S. Sola, E. Levin, M. Fleisher, “Accelerated learning in layered neural networks,” Complex Syst. 2, 625–640 (1988).

IEEE Trans. Inf. Theory (2)

J. Ziv, M. Zakai, “On functionals satisfying a data-processing theorem,”IEEE Trans. Inf. Theory IT-19, 275–283 (1973).
[Crossref]

A. B. VanderLugt, “Signal detection by complex spatial filtering,”IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

J. Opt. Soc. Am. A (3)

Opt. Eng. (1)

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

Opt. Lett. (1)

Other (7)

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

B. V. K. Vijaya Kumar, Z. Bahri, L. Hassebrook, “Review of SDF algorithm,” in Real-Time Signal Processing for Industrial Applications, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.960, 18–29 (1988).

C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (University of Illinois, Urbana, Ill., 1949).

V. Sharma, D. Casasent, “Optimal linear discriminant functions,” in Solid State Imagers and Their Applications, G. J. Declerk, ed., Proc. Soc. Photo-Opt. Instrum. Eng.591, 50–55 (1985).

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

U. Mahlab, J. Rosen, J. Shamir, “Iterative generation of complex RDF in joint transform correlators,” Opt. Lett. (to be published.)

U. Mahlab, J. Shamir, J. H. Caulfield, “Genetic algorithm for optical pattern recognition,” Opt. Lett. (to be published).
[PubMed]

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Figures (4)

Fig. 1
Fig. 1

Input training set: the detected pattern (P) and the pattern to be rejected (F).

Fig. 2
Fig. 2

Output correlation intensity for different filters. The performance of the conventional matched filter to P is presented for comparison in (a), while for each of the others it is based on the noted convex or concave function. For each SDF the discrimination ratio is 4:1.

Fig. 3
Fig. 3

Laboratory experimental result for (a) the log function and, for comparison, (b) a conventional matched filter.

Fig. 4
Fig. 4

Basic configuration of the iterative system: OPs are optical signals, O/E is an optical-to-electronic converter, and ELs are electronic signals.

Tables (1)

Tables Icon

Table 1 Comparison between Several Convex (Concave) Functions in Terms of the Number of Iterations Required to Achieve the Same 4:1 Discrimination Ratio and the Time Used for Each Computer-Simulated Iteration

Equations (59)

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c ( x 0 , y 0 ) = - - f ( x , y ) h * ( x + x 0 , y + y 0 ) d x d y
c ( m , n ) = i = 1 N j = 1 N f ( i , j ) h * ( i + m , j + n ) ,             m , n = 1 , 2 , , ( 2 N - 1 ) ,
φ ( m , n ) = L [ c ( m , n ) ] ,             φ ( m , n ) 0 ,             ( m , n ) .
Φ ( m , n ) = φ ( m , n ) j = 1 2 N - 1 l = 1 2 N - 1 φ ( j , l ) ,
0 Φ ( m , n ) 1 ,             ( m , n ) ,             m , n Φ ( m , n ) = 1 .
min { m = 1 2 N - 1 n = 1 2 N - 1 Ψ [ Φ ( m , n ) ] }
Φ ( m , n ) = constant ,             ( m , n )
max { m = 1 2 N - 1 n = 1 2 N - 1 Ψ [ Φ ( m , n ) ] }
Φ ( m , n ) = { 1 , m = k , n = l , ( k , l ) ( domain of Φ ) 0 , otherwise
Ψ [ Φ ( m , n ) ] < ,             Φ ( m , n ) [ 0 , 1 ] ,             Ψ C 1 ,
S k D = m = 1 2 N - 1 n = 1 2 N - 1 Ψ [ Φ k D ( m , n ) ] ,
S k R = m = 1 2 N - 1 n = 1 2 N - 1 Ψ [ Φ k R ( m , n ) ] ,
M h = { k R } S k R - { k D } S k D
M h = { k D } S k D - { k R } S k R
M ideal = S min R - S max D .
H Ψ = m n Φ ( m , n ) Ψ [ 1 Φ ( m , n ) ] ,
lim α 0 α Ψ ( 1 / α ) = 0 ,
m n Φ ( m , n ) = 1 ,             Φ ( m , n ) 0 ,             ( m , n ) .
Ψ ( α ) = log ( α ) .
H log = - m n Φ ( m , n ) log Φ ( m , n ) .
Ψ 1 ( α ) = log ( α ) ,
Ψ 2 ( α ) = α ,
Ψ 3 ( α ) = ( 1 / α ) - 1 ,
Ψ 4 ( α ) = exp ( - α ) ,
Ψ 5 ( α ) = α p ,             p > 1 ,
Ψ 6 ( α ) = ( max { α } ) 2 .
L { c ( m , n ) } = P { c ( m , n ) } .
φ ( m , n ) = L { c ( m , n ) } = c ( m , n ) .
M h Ψ 1 = m , n Φ R ( m , n ) log Φ R ( m , n ) - m , n Φ D ( m , n ) log Φ D ( m , n ) ,
M h Ψ 2 = m , n [ Φ D ( m , n ) ] 1 / 2 - m , n [ Φ R ( m , n ) ] 1 / 2 ,
M h Ψ 3 = m , n { [ Φ R ( m , n ) ] 2 - Φ R ( m , n ) } - m , n { [ Φ D ( m , n ) ] 2 - Φ D ( m , n ) } ,
M h Ψ 4 = m , n Φ R ( m , n ) exp [ - 1 Φ R ( m , n ) ] - m , n Φ D ( m , n ) exp [ - 1 Φ D ( m , n ) ] ,
M h Ψ 5 = m , n [ Φ R ( m , n ) ] P - m , n [ Φ D ( m , n ) ] P ,             p = 2 ,
M h Ψ 6 = m , n [ max { Φ R ( m , n ) } ] 2 - [ max { Φ D ( m , n ) } ] 2 .
Δ M = M Ψ ( h l + 1 ) - M Ψ ( h l ) .
Pr accept = exp ( - Δ M / T ) ,
M h = m , n [ Φ R ( m , n ) ] 2 - m , n [ Φ D ( m , n ) ] 2 ,
Φ D / R ( m , n ) = c D / R ( m , n ) Σ l , j c D / R ( l , j ) .
M h = ( m , n ) ( 0 , 0 ) Φ R ( m , n ) 2 + ( m , n ) ( 0 , 0 ) Φ D ( m , n ) 2 - τ 1 ( Φ D ( 0 , 0 ) - I 1 ) - τ 2 ( Φ R ( 0 , 0 ) - I 2 ) ,
r = c D ( 0 , 0 ) 2 max ( m , n ) { c R ( m , n ) 2 , c D ( m , n ) ( m , n ) ( 0 , 0 ) 2 } ,
M h = [ max { Φ R ( m , n ) } ] 2 - [ max { Φ D ( m , n ) } ] 2 .
I = Φ D ( 0 , 0 ) 2 - 1 ,
M h = [ max ( m , n ) { Φ R ( m , n ) , Φ D ( m , n ) ( m , n ) ) 0 , 0 ) } ] 2 + τ ( Φ D ( 0 , 0 ) 2 - 1 ) ,
M h general = { k R } S k R { k D } S k D + l N τ l I l ,
H ( U , V ) = 1 ,             ( U , V ) ,
I l = i , j c l ( i , j ) = 1 ,             l = 1 , , N .
Ψ [ λ X 1 + ( 1 - λ ) X 2 ] λ Ψ ( X 1 ) + ( 1 - λ ) Ψ ( X 2 ) ,
Ψ [ λ X 1 + ( 1 - λ ) X 2 ] < λ Ψ ( X 1 ) + ( 1 - λ ) Ψ ( X 2 ) ,
Ψ [ λ X 1 + ( 1 - λ ) X 2 ] λ Ψ ( X 1 ) + ( 1 - λ ) Ψ ( X 2 ) ,
x i 0 ,             i = 1 , 2 , , N
i = 1 N x i = 1
Ψ ( x i ) < ,             x i ,             Ψ C 1 .
min i = 1 N Ψ ( x i ) ,             x i = ( 1 / N ) ,             i ,
max i = 1 N Ψ ( x i ) ,             x i = 1 ,             i = l , x i = 0 ,             i l .
min X { i = 1 N Ψ ( x i ) } ,             max X { i = 1 N Ψ ( x i ) } .
g ( i = 1 N α i x i ) i = 1 N α i g ( x i ) ,
i = 1 N α i = 1 ,             α i 0 ,             ( i ) ,
i = 1 N Ψ ( x i ) = N i = 1 N 1 N Ψ ( x i ) N Ψ [ i = 1 N 1 N ( x i ) ] .
i = 1 N Ψ ( x i ) N Ψ ( 1 N ) .

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