Abstract

A most attractive approach to distortion-invariant pattern recognition uses a synthetic discriminant function (SDF) as the matched spatial filter in a correlator. In this paper, we (1) provide a general basis function and hyperspace description of SDFs, (2) advance a derivation showing the generality of the correlation matrix observation space that we use in our filter synthesis, and (3) detail a unified SDF filter synthesis technique for five different types of pattern recognition problem.

© 1984 Optical Society of America

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References

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  1. A. VanderLugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).
    [CrossRef]
  2. C. Hester, D. Casasent, Proc. Soc. Photo-Opt. Instrum. Eng. 201, 77 (Aug.1979).
  3. C. F. Hester, D. Casasent, Appl. Opt. 19, 1758 (1980).
    [CrossRef] [PubMed]
  4. H. J. Caulfield, Appl. Opt. 19, 3877 (1980).
    [CrossRef] [PubMed]
  5. D. Casasent, V. Sharma, B. V. K. Vijaya Kumar, Proc. Soc. Photo-Opt. Instrum. Eng. 360, 136 (Aug.1982).
  6. J. H. Leger, S. H. Lee, Appl. Opt. 21, 274 (1982).
    [CrossRef] [PubMed]
  7. J. H. Leger, S. H. Lee, J. Opt. Soc. Am. 72, 556 (1982).
    [CrossRef]
  8. H. J. Caulfield, W. T. Maloney, Appl. Opt. 8, 2354 (1969).
    [CrossRef] [PubMed]
  9. R. Duda, P. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973); Y. T. Chien, Interactive Pattern Recognition (Marcel Dekker, New York, 1978).
  10. C. Hester, D. Casasent, Proc. Soc. Photo-Opt. Instrum. Eng. 302, 108 (1981).
  11. C. Hester, D. Casasent, Proc. Soc. Photo-Opt. Instrum. Eng. 292, 25 (1981).
  12. Y-N. Hsu, H. H. Arsenault, Appl. Opt. 21, 4016 (1982).
    [CrossRef] [PubMed]
  13. J. Duvernoy, Opt. Commun. 42, 386 (1982).
    [CrossRef]
  14. J. B. Nelson, Appl. Opt. 20, 8 (1981).
    [CrossRef] [PubMed]
  15. H. Murakami, B. V. K. Vijaya Kumar, IEEE Trans. Pattern. Anal. Machine Intell. PAMI-00, 511 (1982).
    [CrossRef]
  16. B. V. K. Vijaya Kumar, Appl. Opt. 22, 1445 (1983).
    [CrossRef]
  17. D. Casasent, V. Sharma, Proc. Soc. Photo-Opt. Instrum. Eng. 442, 47 (Aug.1983).
  18. J. Flenret, H. Maitre, Opt. Commun. 17, 64 (1976).
    [CrossRef]
  19. B. Braunecker, R. Hauck, A. W. Lohmann, Appl. Opt. 18, 2746 (1979).
    [CrossRef] [PubMed]
  20. D. Casasent, V. Sharma, in Proceedings, International Optical Computing Conference, IEEE Cat. CH1880-4/83 (Apr.1983).

1983 (2)

B. V. K. Vijaya Kumar, Appl. Opt. 22, 1445 (1983).
[CrossRef]

D. Casasent, V. Sharma, Proc. Soc. Photo-Opt. Instrum. Eng. 442, 47 (Aug.1983).

1982 (6)

H. Murakami, B. V. K. Vijaya Kumar, IEEE Trans. Pattern. Anal. Machine Intell. PAMI-00, 511 (1982).
[CrossRef]

Y-N. Hsu, H. H. Arsenault, Appl. Opt. 21, 4016 (1982).
[CrossRef] [PubMed]

J. Duvernoy, Opt. Commun. 42, 386 (1982).
[CrossRef]

D. Casasent, V. Sharma, B. V. K. Vijaya Kumar, Proc. Soc. Photo-Opt. Instrum. Eng. 360, 136 (Aug.1982).

J. H. Leger, S. H. Lee, Appl. Opt. 21, 274 (1982).
[CrossRef] [PubMed]

J. H. Leger, S. H. Lee, J. Opt. Soc. Am. 72, 556 (1982).
[CrossRef]

1981 (3)

J. B. Nelson, Appl. Opt. 20, 8 (1981).
[CrossRef] [PubMed]

C. Hester, D. Casasent, Proc. Soc. Photo-Opt. Instrum. Eng. 302, 108 (1981).

C. Hester, D. Casasent, Proc. Soc. Photo-Opt. Instrum. Eng. 292, 25 (1981).

1980 (2)

1979 (2)

C. Hester, D. Casasent, Proc. Soc. Photo-Opt. Instrum. Eng. 201, 77 (Aug.1979).

B. Braunecker, R. Hauck, A. W. Lohmann, Appl. Opt. 18, 2746 (1979).
[CrossRef] [PubMed]

1976 (1)

J. Flenret, H. Maitre, Opt. Commun. 17, 64 (1976).
[CrossRef]

1969 (1)

1964 (1)

A. VanderLugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).
[CrossRef]

Arsenault, H. H.

Braunecker, B.

Casasent, D.

D. Casasent, V. Sharma, Proc. Soc. Photo-Opt. Instrum. Eng. 442, 47 (Aug.1983).

D. Casasent, V. Sharma, B. V. K. Vijaya Kumar, Proc. Soc. Photo-Opt. Instrum. Eng. 360, 136 (Aug.1982).

C. Hester, D. Casasent, Proc. Soc. Photo-Opt. Instrum. Eng. 302, 108 (1981).

C. Hester, D. Casasent, Proc. Soc. Photo-Opt. Instrum. Eng. 292, 25 (1981).

C. F. Hester, D. Casasent, Appl. Opt. 19, 1758 (1980).
[CrossRef] [PubMed]

C. Hester, D. Casasent, Proc. Soc. Photo-Opt. Instrum. Eng. 201, 77 (Aug.1979).

D. Casasent, V. Sharma, in Proceedings, International Optical Computing Conference, IEEE Cat. CH1880-4/83 (Apr.1983).

Caulfield, H. J.

Duda, R.

R. Duda, P. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973); Y. T. Chien, Interactive Pattern Recognition (Marcel Dekker, New York, 1978).

Duvernoy, J.

J. Duvernoy, Opt. Commun. 42, 386 (1982).
[CrossRef]

Flenret, J.

J. Flenret, H. Maitre, Opt. Commun. 17, 64 (1976).
[CrossRef]

Hart, P.

R. Duda, P. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973); Y. T. Chien, Interactive Pattern Recognition (Marcel Dekker, New York, 1978).

Hauck, R.

Hester, C.

C. Hester, D. Casasent, Proc. Soc. Photo-Opt. Instrum. Eng. 302, 108 (1981).

C. Hester, D. Casasent, Proc. Soc. Photo-Opt. Instrum. Eng. 292, 25 (1981).

C. Hester, D. Casasent, Proc. Soc. Photo-Opt. Instrum. Eng. 201, 77 (Aug.1979).

Hester, C. F.

Hsu, Y-N.

Lee, S. H.

Leger, J. H.

Lohmann, A. W.

Maitre, H.

J. Flenret, H. Maitre, Opt. Commun. 17, 64 (1976).
[CrossRef]

Maloney, W. T.

Murakami, H.

H. Murakami, B. V. K. Vijaya Kumar, IEEE Trans. Pattern. Anal. Machine Intell. PAMI-00, 511 (1982).
[CrossRef]

Nelson, J. B.

Sharma, V.

D. Casasent, V. Sharma, Proc. Soc. Photo-Opt. Instrum. Eng. 442, 47 (Aug.1983).

D. Casasent, V. Sharma, B. V. K. Vijaya Kumar, Proc. Soc. Photo-Opt. Instrum. Eng. 360, 136 (Aug.1982).

D. Casasent, V. Sharma, in Proceedings, International Optical Computing Conference, IEEE Cat. CH1880-4/83 (Apr.1983).

VanderLugt, A.

A. VanderLugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).
[CrossRef]

Vijaya Kumar, B. V. K.

B. V. K. Vijaya Kumar, Appl. Opt. 22, 1445 (1983).
[CrossRef]

D. Casasent, V. Sharma, B. V. K. Vijaya Kumar, Proc. Soc. Photo-Opt. Instrum. Eng. 360, 136 (Aug.1982).

H. Murakami, B. V. K. Vijaya Kumar, IEEE Trans. Pattern. Anal. Machine Intell. PAMI-00, 511 (1982).
[CrossRef]

Appl. Opt. (8)

IEEE Trans. Inf. Theory (1)

A. VanderLugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).
[CrossRef]

IEEE Trans. Pattern. Anal. Machine Intell. (1)

H. Murakami, B. V. K. Vijaya Kumar, IEEE Trans. Pattern. Anal. Machine Intell. PAMI-00, 511 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

J. Duvernoy, Opt. Commun. 42, 386 (1982).
[CrossRef]

J. Flenret, H. Maitre, Opt. Commun. 17, 64 (1976).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (5)

C. Hester, D. Casasent, Proc. Soc. Photo-Opt. Instrum. Eng. 302, 108 (1981).

C. Hester, D. Casasent, Proc. Soc. Photo-Opt. Instrum. Eng. 292, 25 (1981).

D. Casasent, V. Sharma, Proc. Soc. Photo-Opt. Instrum. Eng. 442, 47 (Aug.1983).

C. Hester, D. Casasent, Proc. Soc. Photo-Opt. Instrum. Eng. 201, 77 (Aug.1979).

D. Casasent, V. Sharma, B. V. K. Vijaya Kumar, Proc. Soc. Photo-Opt. Instrum. Eng. 360, 136 (Aug.1982).

Other (2)

R. Duda, P. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973); Y. T. Chien, Interactive Pattern Recognition (Marcel Dekker, New York, 1978).

D. Casasent, V. Sharma, in Proceedings, International Optical Computing Conference, IEEE Cat. CH1880-4/83 (Apr.1983).

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Figures (2)

Fig. 1
Fig. 1

Simplified two-axis hyperspace description of distortion-invariant multiclass shift-invariant pattern recognition using a feature vector and discriminant vector hyperspace concept.

Fig. 2
Fig. 2

Simplified block diagram of the off-line synthetic discriminant function synthesis from training set data and the use of such filters for on-line correlation of the SDF with unknown input imagery.

Tables (1)

Tables Icon

Table I Truth Table for a K-tuple Nonredundant SDF. The Case of M = 4 Classes and K = 2 Filters is Shown.

Equations (36)

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f n ( x , y ) h ( x , y ) = 1.
f n ( x , y ) = m a n m ϕ m ( x , y ) .
h ( x , y ) = m b m ϕ m ( x , y ) .
f n ( x , y ) h ( x , y ) = f n · h = n a n m b m = 1.
h ( x , y ) = b 1 ϕ 1 ( x , y ) + b 2 ϕ 2 ( x , y ) + = m b m ϕ m ( x , y ) .
ϕ m ( x , y ) = n d m n f n ( x , y ) .
h ( x , y ) = b 1 n d 1 n f n ( x , y ) + b 2 d 2 n f n ( x , y ) +
= e 1 f 2 ( x , y ) + e 2 f 2 ( x , y ) +
= m e m f m ( x , y ) .
f n · h = 1.
f n · h = f n · [ m e m f m ] = m e m ( f n · f m ) = m e m r n m = 1.
Re = u ^ ,
e = R - 1 u ^ .
a = R 1 - 1 u ^ 1 = R 1 - 1 [ 1 , , 1 ] T ,
h ( x , y ) = n a n f n ( x , y ) ,
f j ( x , y ) h i ( x , y ) = δ i j ,
h 1 ( x , y ) = m a m f m ( x , y ) ,             h 2 ( x , y ) = m b m f m ( x , y ) , , h M ( x , y ) = m M m f m ( x , y ) ,
h p ( x , y ) = m p m f m ( x , y ) ,
a R 2 = u ^ a = [ 1 , 0 , 0 , 0 , , 0 ] T ,
b R 2 = u ^ b = [ 0 , 1 , 0 , 0 , , 0 ] T ,
c R 2 = u ^ c = [ 0 , 0 , 1 , 0 , , 0 ] T , etc .
a = R 2 - 1 u ^ a ,             b = R 2 - 1 u ^ b ,             c = R 2 - 1 u ^ c , etc .
f n i ( x , y ) h m ( x , y ) = δ n m ,
h a ( x , y ) = n a n f n ( x , y ) , h b ( x , y ) = n b n f n ( x , y ) , h c ( x , y ) = n c n f n ( x , y ) ,
a R 3 = u ^ a = [ 1 , , 1 ; 0 , , 0 ; 0 , , 0 ] T ,
b R 3 = u ^ b = [ 0 , , 0 ; 1 , , 1 ; 0 , , 0 ] T ,
c R 3 = u ^ c = [ 0 , , 0 ; 0 , , 0 ; 1 , , 1 ] T .
a = R 3 - 1 u ^ a ,             b = R 3 - 1 u ^ b ,             c = R 3 - 1 u ^ c
a = R 4 - 1 u ^ a ,
f n ( x , y ) h ( x , y ) = n ,
h ( x , y ) = m a m f m ( x , y ) ,
a = R 3 - 1 u ^ 3 ,
h a ( x , y ) = n a n f n ( x , y ) ,             h b ( x , y ) = n b n f n ( x , y ) ,
[ R ] [ a 1 b 1 a 4 N b 4 N ] = [ 0             0 . 0             1 . 1             0 . 1             1 ]
R [ ab ] = [ u ^ 1 u ^ 2 ] ,
[ ab ] = R - 1 [ u ^ 1 u ^ 2 ] .

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