Abstract

In this paper we consider the problem of detecting a target regardless of its orientation when it is known that the target must be from one of two classes. We assume significant random intraclass variability, a complication which requires techniques from statistical pattern recognition for amelioration. The Foley-Sammon transformation for selecting optimum features from random training samples is used to solve the problem.

© 1985 Optical Society of America

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References

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  1. D. Casasent, D. Psaltis, “Position, Rotation, and Scale Invariant Optical Correlation,” Appl. Opt. 15, 1795 (1976).
    [CrossRef] [PubMed]
  2. The problem of scale invariance was also considered in Ref. 1. We shall consider this issue in a subsequent paper.
  3. Y-N. Hsu, H. H. Arsenault, G. April, “Rotation-Invariant, Digital Pattern Recognition Using Circular Harmonic Expansion,” Appl. Opt. 21, 4012 (1982).
    [CrossRef] [PubMed]
  4. Y-N. Hsu, H. H. Arsenault, “Optical Pattern Recognition Using Circular Harmonic Expansion,” Appl. Opt. 21, 4016 (1982).
    [CrossRef] [PubMed]
  5. H. H. Arsenault, Y-N. Hsu, “Rotation-Invariant Discrimination Between Almost Similar Objects,” Appl. Opt. 22, 130 (1983).
    [CrossRef] [PubMed]
  6. E. G. Paek, S. S. Lee, “Discrimination Enhancement in Optical Pattern Recognition by Using a Modified Matched Filter,” Can. J. Phys. 57, 1335 (1979).
    [CrossRef]
  7. R. Wu, H. Stark, “Rotation-Invariant Pattern Recognition Using a Vector Reference,” Appl. Opt. 23, 838 (1984).
    [CrossRef] [PubMed]
  8. K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972), p. 261.
  9. D. H. Foley, J. W. Sammon, “An Optimal Set of Discriminant Vectors,” IEEE Trans. Comput. C-24, 281 (1975).
    [CrossRef]
  10. H. Stark, D. Lee, “An Optical-Digital Approach to the Pattern Recognition of Coal-Workers' Pneumoconiosis,” IEEE Trans. Syst. Man Cybern. SMC-6, 788 (1976).
  11. R. K. O'Toole, H. Stark, “Comparative Study of Optical-Digital vs All-Digital Techniques in Textural Pattern Recognition,” Appl. Opt. 19, 2496 (1980).
    [CrossRef]
  12. The regions {C} are M-dimensional parallelepipeds whose dimensions along different coordinates are determined from the training samples. Typically each side of this hyperbox is centered at the projected center on that coordinate and has dimension 2σ where σ is the standard deviation of the projected samples along the same coordinate.

1984 (1)

1983 (1)

1982 (2)

1980 (1)

1979 (1)

E. G. Paek, S. S. Lee, “Discrimination Enhancement in Optical Pattern Recognition by Using a Modified Matched Filter,” Can. J. Phys. 57, 1335 (1979).
[CrossRef]

1976 (2)

D. Casasent, D. Psaltis, “Position, Rotation, and Scale Invariant Optical Correlation,” Appl. Opt. 15, 1795 (1976).
[CrossRef] [PubMed]

H. Stark, D. Lee, “An Optical-Digital Approach to the Pattern Recognition of Coal-Workers' Pneumoconiosis,” IEEE Trans. Syst. Man Cybern. SMC-6, 788 (1976).

1975 (1)

D. H. Foley, J. W. Sammon, “An Optimal Set of Discriminant Vectors,” IEEE Trans. Comput. C-24, 281 (1975).
[CrossRef]

April, G.

Arsenault, H. H.

Casasent, D.

Foley, D. H.

D. H. Foley, J. W. Sammon, “An Optimal Set of Discriminant Vectors,” IEEE Trans. Comput. C-24, 281 (1975).
[CrossRef]

Fukunaga, K.

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972), p. 261.

Hsu, Y-N.

Lee, D.

H. Stark, D. Lee, “An Optical-Digital Approach to the Pattern Recognition of Coal-Workers' Pneumoconiosis,” IEEE Trans. Syst. Man Cybern. SMC-6, 788 (1976).

Lee, S. S.

E. G. Paek, S. S. Lee, “Discrimination Enhancement in Optical Pattern Recognition by Using a Modified Matched Filter,” Can. J. Phys. 57, 1335 (1979).
[CrossRef]

O'Toole, R. K.

Paek, E. G.

E. G. Paek, S. S. Lee, “Discrimination Enhancement in Optical Pattern Recognition by Using a Modified Matched Filter,” Can. J. Phys. 57, 1335 (1979).
[CrossRef]

Psaltis, D.

Sammon, J. W.

D. H. Foley, J. W. Sammon, “An Optimal Set of Discriminant Vectors,” IEEE Trans. Comput. C-24, 281 (1975).
[CrossRef]

Stark, H.

Wu, R.

Appl. Opt. (6)

Can. J. Phys. (1)

E. G. Paek, S. S. Lee, “Discrimination Enhancement in Optical Pattern Recognition by Using a Modified Matched Filter,” Can. J. Phys. 57, 1335 (1979).
[CrossRef]

IEEE Trans. Comput. (1)

D. H. Foley, J. W. Sammon, “An Optimal Set of Discriminant Vectors,” IEEE Trans. Comput. C-24, 281 (1975).
[CrossRef]

IEEE Trans. Syst. Man Cybern. (1)

H. Stark, D. Lee, “An Optical-Digital Approach to the Pattern Recognition of Coal-Workers' Pneumoconiosis,” IEEE Trans. Syst. Man Cybern. SMC-6, 788 (1976).

Other (3)

The regions {C} are M-dimensional parallelepipeds whose dimensions along different coordinates are determined from the training samples. Typically each side of this hyperbox is centered at the projected center on that coordinate and has dimension 2σ where σ is the standard deviation of the projected samples along the same coordinate.

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972), p. 261.

The problem of scale invariance was also considered in Ref. 1. We shall consider this issue in a subsequent paper.

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Figures (6)

Fig. 1
Fig. 1

Flow diagram showing sequential logic of the classification algorithm.

Fig. 2
Fig. 2

Intraclass variations realized by randomly varying the font of letter B.

Fig. 3
Fig. 3

Results of the first experiment: (A) the test scene; (B) the reference letters; (C) highlighted pixels showing detected location of center of various B; (D) highlighted pixels showing location of centers of various P.

Fig. 4
Fig. 4

Intraclass variability of genus Raja.

Fig. 5
Fig. 5

Intraclass variability of genus Anchoa.

Fig. 6
Fig. 6

Results of the second experiment: (A) the test scene; (B) the reference fish; (C) highlighted pixels showing detected locations of members of Raja; (D) highlighted pixels showing detected locations of members of Anchoa.

Equations (46)

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f ( r , θ ) = n = f n ( r ) exp ( j n θ ) ,
f n ( r ) = 1 2 π 0 2 π f ( r , θ ) exp ( j n θ ) d θ .
z ( Z 1 , , Z N ) T ,
z n | f n * ( x , y ) f ( x , y ) dxdy | .
z 1 ( Z 11 , , Z 1 N ) T ,
z 2 ( Z 21 , , Z 2 N ) T ,
Z 1 n = | a n * ( x , y ) f ( x , y ) dxdy | , n = 1 , , N ,
Z 2 n = | b n * ( x , y ) f ( x , y ) dxdy | ,
R ( ϕ ) = [ ϕ T ( μ 1 μ 2 ) ] 2 ϕ 2 ( K 1 + K 2 ) ϕ .
Y = AZ ,
μ j = 1 L l = 1 L z l ( i ) j = 1 , 2 ,
K j 1 L l = 1 L [ z l ( j ) μ ( j ) ] [ z l ( j ) μ ( j ) ] T .
| a n * ( x , y ) f ( x u , y υ ) dxdy |
| b n * ( x , y ) f ( x u , y υ ) dxdy |
R ( ϕ ) = ϕ T M ϕ ϕ T K ϕ ,
R ( ϕ 1 ) = max ϕ ϕ T M ϕ ϕ T K ϕ .
R ( ϕ i ) = max ϕ ϕ T M ϕ ϕ T K ϕ
ϕ 1 T ϕ i = = ϕ i 1 T ϕ i = 0 .
Q R ( ϕ ) L 1 ϕ 1 T ϕ L i 1 ϕ i 1 T ϕ ,
ϕ Q = 0
ϕ 1 T ϕ i = 0 , ϕ i 1 T ϕ i = 0 .
[ σ 1 2 + σ 2 2 ] 1 ϕ T M ϕ ϕ 2
R ( ϕ 1 ) = max ϕ ϕ T M ϕ
R ( ϕ i ) = max ϕ ϕ T M ϕ
ϕ 1 T ϕ i = = ϕ i 1 T ϕ i = 0 .
ϕ 1 = μ 1 μ 2 μ 1 μ 2 ,
ϕ Bz .
R ( ϕ ) R ( Bz ) = z T Γ z z 2 = y T Γ z ,
R ( y 1 ) = max y y T Γ y .
R ( y i ) = max y y T Γ y
y 1 T Cy = = y i 1 Cy = 0 ,
Q R ( y ) L 1 y 1 T Cy L i 1 y i 1 T Cy L i y 2 ,
y Q = 0 ,
y j T Cy = 0 , j = 1 , , i 1 ,
y 2 = 1 .
y Q = 0 = 2 Γ y j = 1 i 1 L j C y j 2 L i y = 2 γ y 1 u n j = 1 i 1 L j C y j 2 L i y ,
y = k [ u n CY β ] ,
j = 1 i 1 β j C y j = CY β .
Y T Cy = 0 .
β = S 1 Y T C u n ,
β = S 1 c 11 u ( i 1 ) ,
y i = k [ y 1 c 11 CYS 1 u ( i 1 ) ] .
ϕ i = k [ ϕ 1 c 11 BCYS 1 u ( i 1 ) ] ,
BCY = BB T BY = BB 1 K 1 BY = K 1 BY
BCY = K 1 [ Z 1 1 ϕ 1 | Z 2 1 ϕ 2 | . | Z i 1 1 ϕ i 1 ] .
ϕ i = k i [ ϕ 1 c 11 K 1 ϕ [ ϕ T K 1 ϕ u ( i 1 ) ] .

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