Abstract

The conventional synthetic discriminant functions (SDF’s) determine a filter matched to a linear combination of the available training images such that the resulting cross-correlation output is constant for all training images. We remove the constraint that the filter must be matched to a linear combination of training images and consider a general solution. This general solution is, however, still a linear combination of modified training images. We investigate the effects of noise in input training images and prove that the conventional SDF’s provide minimum output variance when the input noise is white. We provide the design equations for minimum-variance synthetic discriminant functions (MVSDF’s) when the input noise is colored. General expressions are also provided to characterize the loss of optimality when conventional SDF’s are used instead of optimal MVSDF’s.

© 1986 Optical Society of America

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References

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  1. D. Casasent, D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77–84 (1977).
    [CrossRef]
  2. Y. N. Hsu, H. H. Arsenault, “Optical character recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [CrossRef] [PubMed]
  3. Y. Yang, Y. N. Hsu, H. H. Arsenault, “Optimum circular filters and their uses in optical pattern recognition,” Opt. Acta 29, 627–644 (1982).
    [CrossRef]
  4. G. F. Schils, D. W. Sweeney, “Rotationally invariant correlation filtering,” J. Opt. Soc. Am. A 2, 1411–1418 (1985).
    [CrossRef]
  5. D. Casasent, M. Krauss, “Polar camera for space-variant pattern recognition,” Appl. Opt. 17, 1559–1561 (1978).
    [CrossRef] [PubMed]
  6. D. Casasent, “Unified synthetic discriminant function computation formulation,” Appl. Opt. 23, 1620–1627 (1984).
    [CrossRef]
  7. B. V. K. Vijaya Kumar, “Efficient approach for designing linear combination filters,” Appl. Opt. 22, 1445–1448 (1983).
    [CrossRef]
  8. H. J. Caulfield, R. Haimes, J. Horner, “Composite matched filters,” Israel J. Technol. 18, 263–267 (1980).
  9. H. J. Caulfield, M. H. Weinberg, “Computer recognition of 2-D patterns using generalized matched filters,” Appl. Opt. 21, 1699–1704 (1982).
    [CrossRef] [PubMed]
  10. J. Riggins, S. Butler, “Simulation of synthetic discriminant function optical implementation,” Opt. Eng. 23, 721–726 (1984).
    [CrossRef]
  11. B. V. K. Vijaya Kumar, E. Pochapsky, D. Casasent, “Optimally considerations in modified matched spatial filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 519, 85–93 (1984).
  12. B. V. K. Vijaya Kumar, E. Pochapsky, “SNR considerations in modified matched spatial filters,” J. Opt. Soc. Am. A 3, 777–786 (1986).
    [CrossRef]
  13. G. F. Schils, D. W. Sweeney, “Iterative technique for the synthesis of optical-correlation filters,” J. Opt. Soc. Am. A 3, 1433–1442 (1986).
    [CrossRef]
  14. W. T. Chang, D. Casasent, D. Fetterly, “SDF control of correlation plane structure for 3-D object representation and recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 507, 9–18 (1984).
  15. C. R. Rao, Linear Statistical Inference and Its Applications (Wiley, New York, 1973).
    [CrossRef]
  16. R. R. Kallman, “Construction of low noise optical correlation filters,” Appl. Opt. 25, 1032–1033 (1986).
    [CrossRef] [PubMed]
  17. C. W. Helstrom, Probability and Stochastic Processes for Engineers (Macmillan, New York, 1984).
  18. G. W. Stewart, Introduction to Matrix Computations (Academic, New York, 1973).
  19. C. R. Rao, S. K. Mitra, Generalized Inverses of Matrices and Their Applications (Wiley, New York, 1971).
  20. B. V. K. Vijaya Kumar, C. Carroll, “Loss of optimality in cross-correlators,” J. Opt. Soc. Am. A 1, 392–397 (1984).
    [CrossRef]

1986 (3)

1985 (1)

1984 (5)

D. Casasent, “Unified synthetic discriminant function computation formulation,” Appl. Opt. 23, 1620–1627 (1984).
[CrossRef]

B. V. K. Vijaya Kumar, C. Carroll, “Loss of optimality in cross-correlators,” J. Opt. Soc. Am. A 1, 392–397 (1984).
[CrossRef]

W. T. Chang, D. Casasent, D. Fetterly, “SDF control of correlation plane structure for 3-D object representation and recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 507, 9–18 (1984).

J. Riggins, S. Butler, “Simulation of synthetic discriminant function optical implementation,” Opt. Eng. 23, 721–726 (1984).
[CrossRef]

B. V. K. Vijaya Kumar, E. Pochapsky, D. Casasent, “Optimally considerations in modified matched spatial filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 519, 85–93 (1984).

1983 (1)

1982 (3)

1980 (1)

H. J. Caulfield, R. Haimes, J. Horner, “Composite matched filters,” Israel J. Technol. 18, 263–267 (1980).

1978 (1)

1977 (1)

D. Casasent, D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77–84 (1977).
[CrossRef]

Arsenault, H. H.

Y. Yang, Y. N. Hsu, H. H. Arsenault, “Optimum circular filters and their uses in optical pattern recognition,” Opt. Acta 29, 627–644 (1982).
[CrossRef]

Y. N. Hsu, H. H. Arsenault, “Optical character recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
[CrossRef] [PubMed]

Butler, S.

J. Riggins, S. Butler, “Simulation of synthetic discriminant function optical implementation,” Opt. Eng. 23, 721–726 (1984).
[CrossRef]

Carroll, C.

Casasent, D.

B. V. K. Vijaya Kumar, E. Pochapsky, D. Casasent, “Optimally considerations in modified matched spatial filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 519, 85–93 (1984).

W. T. Chang, D. Casasent, D. Fetterly, “SDF control of correlation plane structure for 3-D object representation and recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 507, 9–18 (1984).

D. Casasent, “Unified synthetic discriminant function computation formulation,” Appl. Opt. 23, 1620–1627 (1984).
[CrossRef]

D. Casasent, M. Krauss, “Polar camera for space-variant pattern recognition,” Appl. Opt. 17, 1559–1561 (1978).
[CrossRef] [PubMed]

D. Casasent, D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77–84 (1977).
[CrossRef]

Caulfield, H. J.

H. J. Caulfield, M. H. Weinberg, “Computer recognition of 2-D patterns using generalized matched filters,” Appl. Opt. 21, 1699–1704 (1982).
[CrossRef] [PubMed]

H. J. Caulfield, R. Haimes, J. Horner, “Composite matched filters,” Israel J. Technol. 18, 263–267 (1980).

Chang, W. T.

W. T. Chang, D. Casasent, D. Fetterly, “SDF control of correlation plane structure for 3-D object representation and recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 507, 9–18 (1984).

Fetterly, D.

W. T. Chang, D. Casasent, D. Fetterly, “SDF control of correlation plane structure for 3-D object representation and recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 507, 9–18 (1984).

Haimes, R.

H. J. Caulfield, R. Haimes, J. Horner, “Composite matched filters,” Israel J. Technol. 18, 263–267 (1980).

Helstrom, C. W.

C. W. Helstrom, Probability and Stochastic Processes for Engineers (Macmillan, New York, 1984).

Horner, J.

H. J. Caulfield, R. Haimes, J. Horner, “Composite matched filters,” Israel J. Technol. 18, 263–267 (1980).

Hsu, Y. N.

Y. Yang, Y. N. Hsu, H. H. Arsenault, “Optimum circular filters and their uses in optical pattern recognition,” Opt. Acta 29, 627–644 (1982).
[CrossRef]

Y. N. Hsu, H. H. Arsenault, “Optical character recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
[CrossRef] [PubMed]

Kallman, R. R.

Krauss, M.

Mitra, S. K.

C. R. Rao, S. K. Mitra, Generalized Inverses of Matrices and Their Applications (Wiley, New York, 1971).

Pochapsky, E.

B. V. K. Vijaya Kumar, E. Pochapsky, “SNR considerations in modified matched spatial filters,” J. Opt. Soc. Am. A 3, 777–786 (1986).
[CrossRef]

B. V. K. Vijaya Kumar, E. Pochapsky, D. Casasent, “Optimally considerations in modified matched spatial filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 519, 85–93 (1984).

Psaltis, D.

D. Casasent, D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77–84 (1977).
[CrossRef]

Rao, C. R.

C. R. Rao, Linear Statistical Inference and Its Applications (Wiley, New York, 1973).
[CrossRef]

C. R. Rao, S. K. Mitra, Generalized Inverses of Matrices and Their Applications (Wiley, New York, 1971).

Riggins, J.

J. Riggins, S. Butler, “Simulation of synthetic discriminant function optical implementation,” Opt. Eng. 23, 721–726 (1984).
[CrossRef]

Schils, G. F.

Stewart, G. W.

G. W. Stewart, Introduction to Matrix Computations (Academic, New York, 1973).

Sweeney, D. W.

Vijaya Kumar, B. V. K.

Weinberg, M. H.

Yang, Y.

Y. Yang, Y. N. Hsu, H. H. Arsenault, “Optimum circular filters and their uses in optical pattern recognition,” Opt. Acta 29, 627–644 (1982).
[CrossRef]

Appl. Opt. (6)

Israel J. Technol. (1)

H. J. Caulfield, R. Haimes, J. Horner, “Composite matched filters,” Israel J. Technol. 18, 263–267 (1980).

J. Opt. Soc. Am. A (4)

Opt. Acta (1)

Y. Yang, Y. N. Hsu, H. H. Arsenault, “Optimum circular filters and their uses in optical pattern recognition,” Opt. Acta 29, 627–644 (1982).
[CrossRef]

Opt. Eng. (1)

J. Riggins, S. Butler, “Simulation of synthetic discriminant function optical implementation,” Opt. Eng. 23, 721–726 (1984).
[CrossRef]

Proc. IEEE (1)

D. Casasent, D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77–84 (1977).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

W. T. Chang, D. Casasent, D. Fetterly, “SDF control of correlation plane structure for 3-D object representation and recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 507, 9–18 (1984).

B. V. K. Vijaya Kumar, E. Pochapsky, D. Casasent, “Optimally considerations in modified matched spatial filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 519, 85–93 (1984).

Other (4)

C. R. Rao, Linear Statistical Inference and Its Applications (Wiley, New York, 1973).
[CrossRef]

C. W. Helstrom, Probability and Stochastic Processes for Engineers (Macmillan, New York, 1984).

G. W. Stewart, Introduction to Matrix Computations (Academic, New York, 1973).

C. R. Rao, S. K. Mitra, Generalized Inverses of Matrices and Their Applications (Wiley, New York, 1971).

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Equations (30)

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h = i = 1 N a i x i ,
h T x j = u j , j = 1 , 2 , , N ,
i = 1 N a i ( x i T x j ) = u j , j = 1 , 2 , , N .
Ra = u ,
h = Xa ,
( X T X ) a = u .
| h T x j | 2 = | u j | 2 , j = 1 , 2 , , N .
E { y } = 0
Var { y } = E { h T nn T h } = h T E { nn T } h = h T Ch ,
ϕ = h T Ch 2 λ 1 ( h T x 1 u 1 ) 2 λ N ( h T x N u N ) ,
Ch opt = λ 1 x 1 + + λ N x N ,
h opt = C 1 [ i = 1 N λ i x i ] = i = 1 N λ i C 1 x i
i = 1 N λ i x j T C 1 x i = u j , j = 1 , 2 , , N .
h opt = C 1 X 1 ,
X T C 1 X 1 = u ,
h R T x i = u i
h I T x i = 0 , i = 1 , 2 , , N .
E { | y | 2 } = E { ( h R T n ) 2 + ( h I T n ) 2 } = h R T Ch R + h I T Ch I .
g = X ( 1 σ 2 1 ) ,
( X T X ) ( 1 σ 2 1 ) = u .
σ opt 2 = h opt T C h opt = ( C 1 X 1 ) T C ( C 1 X 1 ) = 1 T X T C 1 X 1 ,
σ opt 2 = [ ( X T C 1 X ) 1 u ] T ( X T C 1 X ) [ ( X T C 1 X ) 1 u ] = u T ( X T C 1 X ) 1 ( X T C 1 X ) ( X T C 1 X ) 1 u = u T ( X T C 1 X ) 1 u .
σ SDF 2 = g T Cg = 1 σ 4 1 T X T CX 1 .
σ SDF 2 = [ ( X T X ) 1 u ] T X T C X [ ( X T X ) 1 u ] = u T ( X T X ) 1 ( X T C X ) ( X T X ) 1 u .
F = u T ( X T C 1 X ) 1 u u T ( X T X ) 1 ( X T CX ) ( X T X ) 1 u .
F = u T ( X T C 1 X ) 1 u u T ( X + CX + T ) u ,
F = ( x T x ) 2 ( x T Cx ) ( x T C 1 x ) ,
F = u T ( X T C 1 X ) 1 u u T ( X T C X ) u .
X T CX = Diag ( α j 1 , , α j N )
( X T CX 1 X ) 1 = Diag ( α j 1 , , α j N ) .

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