Abstract

Circular harmonic filters of different orders are compared using a criterion of separability between the target and object output probability densities. A holographic computer-generated circular harmonic filter was used to recognize aircraft from an air photograph.

© 1983 Optical Society of America

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References

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  1. A. B. VanderLugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).
  2. Y.-N. Hsu, H. H. Arsenault, G. April, Appl. Opt. 21, 4012 (1982).
    [CrossRef] [PubMed]
  3. Y.-N. Hsu, H. H. Arsenault, Appl. Opt. 21, 4016 (1982).
    [CrossRef] [PubMed]
  4. H. H. Arsenault, Y.-N. Hsu, K. Chalasinska-Macukow, Y. Yang, Proc. Soc. Photo-Opt. Instrum. Eng. 359, 266 (1982).
  5. W. S. Meisel, Computer-Oriented Approaches to Pattern Recognition (Academic, New York, 1972).
  6. J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle, J. C. Dainty, Ed. (Springer, New York, 1975), p. 15.
  7. Ref. 6, pp. 29-33.
  8. A. Bhattacharyya, Bull. Calcutta Math. Soc. 35, 99 (1943).
  9. S. Lowenthal, P. Chavel, Appl. Opt. 13, 718 (1974).
    [CrossRef] [PubMed]

1982 (3)

Y.-N. Hsu, H. H. Arsenault, G. April, Appl. Opt. 21, 4012 (1982).
[CrossRef] [PubMed]

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

H. H. Arsenault, Y.-N. Hsu, K. Chalasinska-Macukow, Y. Yang, Proc. Soc. Photo-Opt. Instrum. Eng. 359, 266 (1982).

1974 (1)

1964 (1)

A. B. VanderLugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).

1943 (1)

A. Bhattacharyya, Bull. Calcutta Math. Soc. 35, 99 (1943).

April, G.

Arsenault, H. H.

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

H. H. Arsenault, Y.-N. Hsu, K. Chalasinska-Macukow, Y. Yang, Proc. Soc. Photo-Opt. Instrum. Eng. 359, 266 (1982).

Y.-N. Hsu, H. H. Arsenault, G. April, Appl. Opt. 21, 4012 (1982).
[CrossRef] [PubMed]

Bhattacharyya, A.

A. Bhattacharyya, Bull. Calcutta Math. Soc. 35, 99 (1943).

Chalasinska-Macukow, K.

H. H. Arsenault, Y.-N. Hsu, K. Chalasinska-Macukow, Y. Yang, Proc. Soc. Photo-Opt. Instrum. Eng. 359, 266 (1982).

Chavel, P.

Goodman, J. W.

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle, J. C. Dainty, Ed. (Springer, New York, 1975), p. 15.

Hsu, Y.-N.

H. H. Arsenault, Y.-N. Hsu, K. Chalasinska-Macukow, Y. Yang, Proc. Soc. Photo-Opt. Instrum. Eng. 359, 266 (1982).

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

Y.-N. Hsu, H. H. Arsenault, G. April, Appl. Opt. 21, 4012 (1982).
[CrossRef] [PubMed]

Lowenthal, S.

Meisel, W. S.

W. S. Meisel, Computer-Oriented Approaches to Pattern Recognition (Academic, New York, 1972).

VanderLugt, A. B.

A. B. VanderLugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).

Yang, Y.

H. H. Arsenault, Y.-N. Hsu, K. Chalasinska-Macukow, Y. Yang, Proc. Soc. Photo-Opt. Instrum. Eng. 359, 266 (1982).

Appl. Opt. (3)

Bull. Calcutta Math. Soc. (1)

A. Bhattacharyya, Bull. Calcutta Math. Soc. 35, 99 (1943).

IEEE Trans. Inf. Theory (1)

A. B. VanderLugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

H. H. Arsenault, Y.-N. Hsu, K. Chalasinska-Macukow, Y. Yang, Proc. Soc. Photo-Opt. Instrum. Eng. 359, 266 (1982).

Other (3)

W. S. Meisel, Computer-Oriented Approaches to Pattern Recognition (Academic, New York, 1972).

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle, J. C. Dainty, Ed. (Springer, New York, 1975), p. 15.

Ref. 6, pp. 29-33.

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Figures (10)

Fig. 1
Fig. 1

Universal curve for the circular harmonic filter figure of merit.

Fig. 2
Fig. 2

Section of an aerial photograph from T. E. Avery, Interpretation of Aerial Photographs (Burgess, Minneapolis, 1968), 2nd edition p. 306.

Fig. 3
Fig. 3

Figures of merit for filters of different orders. The target is shown in Fig. 4(a) with size rm. □, Figures of merit calculated for the discrimination between the target and a white square with a side equal to rm; +, Figures of merit calculated for the discrimination between the target and road intersections 0.65 rm wide.

Fig. 4
Fig. 4

Circular harmonic component of the airplane pattern: (a) standard pattern; the proper center is denoted by ×; (b) modulus of the circular harmonic component (M = 6); (c) real part of the circular harmonic component; (d) imaginary part of the circular harmonic component.

Fig. 5
Fig. 5

Computer-generated binary filter corresponding to the circular harmonic component of Fig. 4.

Fig. 6
Fig. 6

Optical system for holographic copying.

Fig. 7
Fig. 7

Diffraction orders at the Fourier plane of the copying system.

Fig. 8
Fig. 8

Reconstruction of the filter image from the holographic copy.

Fig. 9
Fig. 9

Output of the optical recognition system.

Fig. 10
Fig. 10

Threshold output of the optical recognition system.

Tables (1)

Tables Icon

Table I Complex Amplitude of the Circular Harmonic Component of Fig. 4

Equations (35)

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f ( r , θ ) = m = f M ( r ) exp ( j M θ )
f M ( r ) = 1 2 π 0 2 π f ( r , θ ) exp ( j M θ ) d θ .
H ( f x , f y ) = * [ f r 1 ( x , y ) ] ,
f R ( r , θ ) = f M ( r ) exp ( j M θ ) .
f M ( r ) = | f M ( r ) | exp [ j ϕ M ( r ) ] .
A ( M ) = 2 π 0 r | f M ( r ) | 2 d r
W ( f x , f y ) = N in 4 rect ( 2 F x ) rect ( 2 F y ) ,
R in ( Δ x , Δ y ) = W ( f x , f y ) × exp [ j 2 π ( f x Δ x + f y Δ y ) ] d f x d f y
R in ( Δ x , Δ y ) N in 4 δ ( Δ x , Δ y )
n out ( x , y ) = n in ( ξ , η ) f R 1 * ( ξ x , η y ) d ξ d η ,
f R 1 ( x , y ) = b r 1 ( x , y ) + j b i 1 ( x , y )
n out ( x , y ) = n r ( x , y ) + j n i ( x , y )
n r ( x , y ) = n in ( ξ , η ) b r 1 ( ξ x , η y ) d ξ d η ,
n i ( x , y ) = n in ( ξ , η ) b i 1 ( ξ x , η y ) d ξ d η ,
R r i ( x , y ) = E [ n r ( x 1 , y 1 ) n i ( x 2 , y 2 ) ] .
R r i ( 0 , 0 ) = N in 4 b r 1 ( α , β ) b i 1 ( α , β ) d α d β .
b r ( r , θ ) = | f M ( r ) | cos [ ϕ M ( r ) + M θ ] ,
b i ( r , θ ) = | f M ( r ) | sin [ ϕ M ( r ) + M θ ] ,
R r i ( 0 , 0 ) = N in 4 0 r d r 0 2 π b r ( r , θ ) b i ( r , θ ) d θ = 0 .
σ 2 = R r r ( 0 , 0 ) = 1 8 N in A ( M ) ,
σ 2 = R i i ( 0 , 0 ) = 1 8 N in A ( M ) .
SNR = A 2 ( M ) 2 σ 2 = 4 A ( M ) N in .
z = | S | exp ( j ψ ) + n r + j n i ,
p ( | z | ) = z σ 2 exp [ 1 2 σ 2 ( | z | 2 + | S | 2 ) I 0 ( | S | | z | σ 2 ) ] ,
I 0 ( x ) = n = 0 x 2 n 2 2 n ( n ! ) 2 .
i = I A 2 ( M ) ,
I = | z | 2 .
p ( i ) = A 2 ( M ) 2 σ 2 exp { 1 2 σ 2 [ A 2 ( M ) i + | S | 2 ] } I 0 [ | S | A ( M ) i σ 2 ] .
p t ( i ) = ( SNR ) exp [ ( SNR ) ( i + 1 ) ] I 0 [ 2 ( SNR ) i ] .
p b ( i ) = ( SNR ) exp { ( SNR ) × [ i + B 2 ( M ) A 2 ( M ) ] } I 0 [ 2 ( SNR ) B ( M ) A ( M ) i ] .
Q = log X [ p 1 ( x ) p 2 ( x ) ] 1 / 2 d x .
Q = { 0 , for A ( M ) B ( M ) , 1 0 p t ( i ) p b ( i ) d i , for A ( M ) > B ( M ) .
P t p t ( i ) P b p b ( i ) d i ,
Q = 1 0 p t ( | z | ) p b ( | z | ) d | z | , for A ( M ) > B ( M ) ,
Q = { 0 , for A ( M ) B ( M ) , 1 ( SNR ) exp [ ( SNR ) 2 ( 1 + B 2 A 2 ) ] 0 { I 0 [ 2 ( SNR ) i ] × I 0 [ 2 ( SNR ) B A i ] } 1 / 2 × exp [ ( SNR ) i ] d i , for A ( M ) > B ( M ) .

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