Abstract

The matched filter (MF) is the optimum linear operator for distinguishing between a fixed signal and noise, given the noise statistics. A generalized matched filter (GMF) is a linear filter that can handle the more difficult problem of a multiple-example signal set, and it reduces to a MF when the signal set has only one member. A supergeneralized matched filter (SGMF) is a set of GMFs and a procedure to combine their results nonlinearly to handle the multisignal problem even better. Obviously the SGMF contains the GMF as a special case. An algorithm for training SGMFs is presented, and it is shown that the algorithm performs quite well even for extremely difficult classification problems.

© 2005 Optical Society of America

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References

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  1. A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
  2. H. J. Caulfield, R. Haimes, D. Casasent, “Beyond matched filtering,” Opt. Eng. 19, 152–156 (1980).
    [CrossRef]
  3. M. A. G. Abushagur, H. J. Caulfield, eds., Selected Papers on Fourier Optics, Vol. MS105 of SPIE Milestone Series (SPIE, Bellingham, Wash., 1995).
  4. H. J. Caulfield, R. Haimes, “Generalized matched filtering,” Appl. Opt. 19, 181–183 (1980).
    [CrossRef] [PubMed]
  5. H. J. Caulfield, M. H. Weinberg, “Computer recognition of 2-D patterns using generalized matched filters,” Appl. Opt. 21, 1699–1704 (1982).
    [CrossRef] [PubMed]
  6. E. R. Dougherty, J. Astola, Nonlinear Filters for Image Processing (Wiley, New York, 1999).
  7. M. H. A. Davis, S. I. Marcus, “An introduction to nonlinear filtering,” in Stochastic Systems: The Mathematics of Filtering and Identification and Applications, M. Hzewinkel, J. C. Willems, eds. (Reidel, Dordrecht, The Netherlands, 1981), pp. 53–75.
  8. S. Lototsky, R. Mikulevicius, B. L. Rozovski, “Nonlinear filtering revisited: a spectral approach,” SIAM J. Control 35, 435–461 (1997).
    [CrossRef]
  9. S. Haykin, Communications Systems, 4th ed. (Wiley, New York, 2001).
  10. H. J. Caulfield, K. Heidary, “Exploring margin setting for good generalization in multiple class discrimination,” J. Pattern Recogn. (to be published).
  11. H. J. Caulfield, A. Karavolos, E. Ludman, “Improving Fourier recognition by accommodating the missing information,” Inf. Sci. (N.Y.) 162, 35–52 (2004).
    [CrossRef]
  12. J. A. Hanley, “Receiver operating characteristic (ROC) curves,” in Encyclopedia of Biostatistics, P. Armitage, T. Colton, eds. (Wiley, New York, 1998), pp. 3738–3745.

2004

H. J. Caulfield, A. Karavolos, E. Ludman, “Improving Fourier recognition by accommodating the missing information,” Inf. Sci. (N.Y.) 162, 35–52 (2004).
[CrossRef]

1997

S. Lototsky, R. Mikulevicius, B. L. Rozovski, “Nonlinear filtering revisited: a spectral approach,” SIAM J. Control 35, 435–461 (1997).
[CrossRef]

1982

1980

H. J. Caulfield, R. Haimes, “Generalized matched filtering,” Appl. Opt. 19, 181–183 (1980).
[CrossRef] [PubMed]

H. J. Caulfield, R. Haimes, D. Casasent, “Beyond matched filtering,” Opt. Eng. 19, 152–156 (1980).
[CrossRef]

1964

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

Astola, J.

E. R. Dougherty, J. Astola, Nonlinear Filters for Image Processing (Wiley, New York, 1999).

Casasent, D.

H. J. Caulfield, R. Haimes, D. Casasent, “Beyond matched filtering,” Opt. Eng. 19, 152–156 (1980).
[CrossRef]

Caulfield, H. J.

H. J. Caulfield, A. Karavolos, E. Ludman, “Improving Fourier recognition by accommodating the missing information,” Inf. Sci. (N.Y.) 162, 35–52 (2004).
[CrossRef]

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, D. Casasent, “Beyond matched filtering,” Opt. Eng. 19, 152–156 (1980).
[CrossRef]

H. J. Caulfield, R. Haimes, “Generalized matched filtering,” Appl. Opt. 19, 181–183 (1980).
[CrossRef] [PubMed]

H. J. Caulfield, K. Heidary, “Exploring margin setting for good generalization in multiple class discrimination,” J. Pattern Recogn. (to be published).

Davis, M. H. A.

M. H. A. Davis, S. I. Marcus, “An introduction to nonlinear filtering,” in Stochastic Systems: The Mathematics of Filtering and Identification and Applications, M. Hzewinkel, J. C. Willems, eds. (Reidel, Dordrecht, The Netherlands, 1981), pp. 53–75.

Dougherty, E. R.

E. R. Dougherty, J. Astola, Nonlinear Filters for Image Processing (Wiley, New York, 1999).

Haimes, R.

H. J. Caulfield, R. Haimes, “Generalized matched filtering,” Appl. Opt. 19, 181–183 (1980).
[CrossRef] [PubMed]

H. J. Caulfield, R. Haimes, D. Casasent, “Beyond matched filtering,” Opt. Eng. 19, 152–156 (1980).
[CrossRef]

Hanley, J. A.

J. A. Hanley, “Receiver operating characteristic (ROC) curves,” in Encyclopedia of Biostatistics, P. Armitage, T. Colton, eds. (Wiley, New York, 1998), pp. 3738–3745.

Haykin, S.

S. Haykin, Communications Systems, 4th ed. (Wiley, New York, 2001).

Heidary, K.

H. J. Caulfield, K. Heidary, “Exploring margin setting for good generalization in multiple class discrimination,” J. Pattern Recogn. (to be published).

Karavolos, A.

H. J. Caulfield, A. Karavolos, E. Ludman, “Improving Fourier recognition by accommodating the missing information,” Inf. Sci. (N.Y.) 162, 35–52 (2004).
[CrossRef]

Lototsky, S.

S. Lototsky, R. Mikulevicius, B. L. Rozovski, “Nonlinear filtering revisited: a spectral approach,” SIAM J. Control 35, 435–461 (1997).
[CrossRef]

Ludman, E.

H. J. Caulfield, A. Karavolos, E. Ludman, “Improving Fourier recognition by accommodating the missing information,” Inf. Sci. (N.Y.) 162, 35–52 (2004).
[CrossRef]

Marcus, S. I.

M. H. A. Davis, S. I. Marcus, “An introduction to nonlinear filtering,” in Stochastic Systems: The Mathematics of Filtering and Identification and Applications, M. Hzewinkel, J. C. Willems, eds. (Reidel, Dordrecht, The Netherlands, 1981), pp. 53–75.

Mikulevicius, R.

S. Lototsky, R. Mikulevicius, B. L. Rozovski, “Nonlinear filtering revisited: a spectral approach,” SIAM J. Control 35, 435–461 (1997).
[CrossRef]

Rozovski, B. L.

S. Lototsky, R. Mikulevicius, B. L. Rozovski, “Nonlinear filtering revisited: a spectral approach,” SIAM J. Control 35, 435–461 (1997).
[CrossRef]

VanderLugt, A.

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

Weinberg, M. H.

Appl. Opt.

IEEE Trans. Inf. Theory

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

Inf. Sci. (N.Y.)

H. J. Caulfield, A. Karavolos, E. Ludman, “Improving Fourier recognition by accommodating the missing information,” Inf. Sci. (N.Y.) 162, 35–52 (2004).
[CrossRef]

Opt. Eng.

H. J. Caulfield, R. Haimes, D. Casasent, “Beyond matched filtering,” Opt. Eng. 19, 152–156 (1980).
[CrossRef]

SIAM J. Control

S. Lototsky, R. Mikulevicius, B. L. Rozovski, “Nonlinear filtering revisited: a spectral approach,” SIAM J. Control 35, 435–461 (1997).
[CrossRef]

Other

S. Haykin, Communications Systems, 4th ed. (Wiley, New York, 2001).

H. J. Caulfield, K. Heidary, “Exploring margin setting for good generalization in multiple class discrimination,” J. Pattern Recogn. (to be published).

J. A. Hanley, “Receiver operating characteristic (ROC) curves,” in Encyclopedia of Biostatistics, P. Armitage, T. Colton, eds. (Wiley, New York, 1998), pp. 3738–3745.

M. A. G. Abushagur, H. J. Caulfield, eds., Selected Papers on Fourier Optics, Vol. MS105 of SPIE Milestone Series (SPIE, Bellingham, Wash., 1995).

E. R. Dougherty, J. Astola, Nonlinear Filters for Image Processing (Wiley, New York, 1999).

M. H. A. Davis, S. I. Marcus, “An introduction to nonlinear filtering,” in Stochastic Systems: The Mathematics of Filtering and Identification and Applications, M. Hzewinkel, J. C. Willems, eds. (Reidel, Dordrecht, The Netherlands, 1981), pp. 53–75.

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Figures (8)

Fig. 1
Fig. 1

Filter design for the two-class problem.

Fig. 2
Fig. 2

Effect of margin and zero threshold on the false-negative error rate for 25 trainers and 2000 test images with set 10 ≤ r ≤ 20, 0 ≤ ϕ ≤ 60° for training and test images.

Fig. 3
Fig. 3

Effect of margin and zero threshold on the false-positive error rate for 25 trainers and 2000 test images with set 10 ≤ r ≤ 20, 0 ≤ ϕ ≤ 60° for training and test images.

Fig. 4
Fig. 4

Effect of margin and zero threshold on the false-negative error rate for 200 trainers and 4000 test images with set 10 ≤ r ≤ 50, 0 ≤ ϕ ≤ 60° for training and test images.

Fig. 5
Fig. 5

Effect of margin and zero threshold on the false-positive error rate for 200 trainers and 4000 test images with set 10 ≤ r ≤ 50, 0 ≤ ϕ ≤ 60° for training and test images.

Fig. 6
Fig. 6

Effect of margin and zero threshold on classification error for 200 trainers and 4000 test images with set 5 ≤ r ≤ 50, 0 ≤ ϕ ≤ 120° for training and test images.

Fig. 7
Fig. 7

ROC for the classifier. Parametric behavior of probability of detection and failure as functions of margin (β) with fixed zero-threshold (V0 = 0.7) and various radius (R) values for 49 trainers and 4000 test images and set 10 ≤ r ≤ 30, 0 ≤ ϕ ≤ 60°.

Fig. 8
Fig. 8

ROC for the classifier. Parametric behavior of probability of detection and failure as functions of margin (β) with fixed radius (R = 0.3) and various zero-threshold (V0 = 0.65, 0.7, 0.75) values for 49 trainers and 4000 test images and set 10 < r < 30, 0 < ϕ < 60°.

Tables (3)

Tables Icon

Table 1 Number of Classifiers (GMFs) in the SGMF Set (Rounds of Classification)

Tables Icon

Table 2 False-Negative-Rate (Percent Missed Target)

Tables Icon

Table 3 False-Positive-Rate (Percent Mislabeled)

Equations (30)

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T = { τ i : 1 i N T } ,
t i = τ i [ m = 0 M - 1 n = 0 N - 1 τ i 2 ( m , n ) ] 1 / 2
T = { t i : 1 i N T } ,
T = { T i : 1 i N T } ,
T i ( p , q ) = m = 0 M - 1 n = 0 N - 1 t i ( m , n ) exp [ - j 2 π ( m p M + n q N ) ] .
T * = { T i * : 1 i N T } ,
λ i , j ( m , n ) = m = 0 M - 1 n = 0 N - 1 { t i ( m , n ) t j [ ( m - m ) , ( n - n ) ] } ,
k , l Z ; t i [ ( m + k M ) , ( n + l N ) ] = t i ( m , n ) .
C = [ c i j ] , 1 i , j N T ,
c i j = max m , n [ λ i j ( m , n ) ] .
λ i , j ( m , n ) = 1 M N p = 0 M - 1 q = 0 N - 1 T i ( p , q ) T j * ( p , q ) × exp [ j 2 π ( m p M + n q N ) ] .
c i , j = λ i , j ( 0 , 0 ) = 1 M N p = 0 M - 1 q = 0 N - 1 T i ( p , q ) T j * ( p , q ) .
R [ 0     1 ] ,
O = { O i : 1 i N 0 } , O T ,
t k O : t l O : c k , l 1 - R , 1 k , l N 0 , N 0 N T .
H k ( p , q ) = i = 1 N 0 α i , k O i * ( p , q ) , α i , k [ - 1 , 1 ] , 1 k K .
H k ( p , q ) = H k ( p , q ) [ p = 0 M - 1 q = 0 N - 1 H k ( p , q ) 2 ] 1 / 2 .
x k , i ( m , n ) = 1 M N p = 0 M - 1 q = 0 N - 1 H k ( p , q ) O i ( p , q ) × exp [ j 2 π ( m p M + n q N ) ] ,
r k , i o = max m , n [ x k , i ( m , n ) ] = 1 M N p = 0 M - 1 q = 0 N - 1 × H k ( p , q ) O i ( p , q ) ,
A k = ( α 1 , k , α 2 , k , , α N 0 , k ) , 1 k K , α i , k [ - 1     1 ] .
A k = ( α 1 , k , α 2 , k , , α N 0 , k ) ,
α i , k ~ N ( α i , k , σ 2 ) .
V = V 0 + 0.01 β ( 1 - V 0 ) , β [ 0     100 ] ,
H = { H k , 1 k K } .
H k ( p , q ) = i = 1 N k γ i , k E i , k * ( p , q ) ,             γ i , k [ - 1     1 ] , 1 k K ,
E k = { E i , k , 1 i N k } , E k T ,
D = O ʀ { D k , 1 k K } .
x k ( m , n ) = 1 M N p = 0 M - 1 q = 0 N - 1 H k ( p , q ) I ( p , q ) × exp [ j 2 π ( m p M + n q N ) ] ,
r k o = max m , n [ x k ( m , n ) ] = 1 M N p = 0 M - 1 q = 0 N - 1 H k ( p , q ) I ( p , q ) ,
r k o V D k = 1 ,

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