Abstract

The use of nonlinear techniques in the Fourier plane of pattern-recognition correlators can improve the correlators' performance in terms of discrimination against objects similar to the target object, correlation-peak sharpness, and correlation noise robustness. Additionally, filter designs have been proposed that provide the linear correlator with invariance properties with respect to input-signal distortions and rotations. We propose simple modifications to presently known distortion-invariant correlator filters that enable these filter designs to be used in a nonlinear correlator architecture. These Fourier-plane nonlinear filters can be implemented electronically, or they may be implemented optically with a nonlinear joint transform correlator. Extensive simulation results are presented that illustrate the performance enhancements that are gained by the unification of nonlinear techniques with these filter designs.

© 1996 Optical Society of America

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  1. A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE. Trans. Inf. Theory IT-10, 139–145 (1964).
    [CrossRef]
  2. C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  3. D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
    [CrossRef]
  4. B. V. K. Vijaya Kumar, “A tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
    [CrossRef]
  5. A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef] [PubMed]
  6. B. V. K. Vijaya Kumar, “Minimum variance synthetic discriminant functions,” J. Opt. Soc. Am. A 3, 1579–1584 (1986).
    [CrossRef]
  7. Ph. Refregier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with the Wiener approach,” Opt. Comput. Process. 1, 245–266 (1991).
  8. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [CrossRef] [PubMed]
  9. A. VanderLugt, F. B. Rotz, “The use of film nonlinearities in optical spatial filtering,” Appl. Opt. 1, 215–222 (1970).
    [CrossRef]
  10. Ph. Refregier, V. Laude, B. Javidi, “Nonlinear joint-transform correlation: an optimal solution for adaptive image discrimination and input noise robustness,” Opt. Lett. 19, 405–407 (1994).
    [PubMed]
  11. B. Javidi, Q. Tang, D. A. Gregory, T. D. Hudson, “Experiments on nonlinear joint transform correlator using an optically addressed spatial light modulator in the Fourier plane,” Appl. Opt. 30, 1772–1776 (1991).
    [CrossRef] [PubMed]
  12. C. F. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
    [CrossRef] [PubMed]
  13. A. Mahalanobis, D. P. Casasent, “Performance evaluation of minimum average correlation energy filters,” Appl. Opt. 30, 561–572 (1991).
    [CrossRef] [PubMed]
  14. B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
    [CrossRef]
  15. B. Javidi, “Generalization of the linear matched filter concept to nonlinear matched filters,” Appl. Opt. 29, 1215–1224 (1990).
    [CrossRef] [PubMed]
  16. K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
    [CrossRef]
  17. W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
    [CrossRef]
  18. Q. Tang, B. Javidi, “Sensitivity of the nonlinear joint transform correlator: experimental investigations,” Appl. Opt. 31, 4016–4024 (1992).
    [CrossRef] [PubMed]
  19. J. L. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992).
    [CrossRef] [PubMed]
  20. W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).
  21. B. Javidi, J. Wang, “Limitation of the classic definition of the correlation signal-to-noise ratio in optical pattern recognition with disjoint signal and scene noise,” Appl. Opt. 31, 6826–6829 (1992).
    [CrossRef] [PubMed]
  22. Ph. Refregier, B. Javidi, G. Zhang, “Minimum mean-square-error filter for pattern recognition with spatially disjoint signal and scene noise,” Opt. Lett. 18, 1453–1455 (1993).
    [CrossRef] [PubMed]
  23. B. Javidi, P. Refregier, P. Willett, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1660–1662 (1993).
    [CrossRef] [PubMed]
  24. D. Casasent, G. Ravichandran, “Advanced distortion-invariant minimum average correlation energy (MACE) filters,” Appl. Opt. 31, 1109–1116 (1992).
    [CrossRef] [PubMed]
  25. G. Ravichandran, D. Casasent, “Minimum noise and correlation energy optical correlation filter,” Appl. Opt. 31, 1823–1833 (1992).
    [CrossRef] [PubMed]
  26. B. Javidi, J. Wang, A. H. Fazlollahi, “Performance of the nonlinear joint transform correlator for images with low-pass characteristics,” Appl. Opt. 33, 834–848 (1994).
    [CrossRef] [PubMed]
  27. D. Painchaud, “Distortion invariant nonlinear optical signal processing,” M.S. thesis (University of Connecticut, Storrs, Conn., 1994).

1994 (2)

1993 (2)

1992 (7)

1991 (3)

1990 (3)

1989 (2)

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
[CrossRef] [PubMed]

1987 (1)

1986 (1)

1980 (1)

1970 (1)

A. VanderLugt, F. B. Rotz, “The use of film nonlinearities in optical spatial filtering,” Appl. Opt. 1, 215–222 (1970).
[CrossRef]

1966 (1)

1964 (1)

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

Casasent, D.

Casasent, D. P.

Fazlollahi, A. H.

Fielding, K. H.

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

Figue, J.

Ph. Refregier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with the Wiener approach,” Opt. Comput. Process. 1, 245–266 (1991).

Flannery, D. L.

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

Goodman, J. W.

Gregory, D. A.

Hahn, W. B.

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

Hassebrook, L.

Hester, C. F.

Horner, J. L.

J. L. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992).
[CrossRef] [PubMed]

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

Hudson, T. D.

Javidi, B.

Ph. Refregier, V. Laude, B. Javidi, “Nonlinear joint-transform correlation: an optimal solution for adaptive image discrimination and input noise robustness,” Opt. Lett. 19, 405–407 (1994).
[PubMed]

B. Javidi, J. Wang, A. H. Fazlollahi, “Performance of the nonlinear joint transform correlator for images with low-pass characteristics,” Appl. Opt. 33, 834–848 (1994).
[CrossRef] [PubMed]

Ph. Refregier, B. Javidi, G. Zhang, “Minimum mean-square-error filter for pattern recognition with spatially disjoint signal and scene noise,” Opt. Lett. 18, 1453–1455 (1993).
[CrossRef] [PubMed]

B. Javidi, P. Refregier, P. Willett, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1660–1662 (1993).
[CrossRef] [PubMed]

B. Javidi, J. Wang, “Limitation of the classic definition of the correlation signal-to-noise ratio in optical pattern recognition with disjoint signal and scene noise,” Appl. Opt. 31, 6826–6829 (1992).
[CrossRef] [PubMed]

Q. Tang, B. Javidi, “Sensitivity of the nonlinear joint transform correlator: experimental investigations,” Appl. Opt. 31, 4016–4024 (1992).
[CrossRef] [PubMed]

B. Javidi, Q. Tang, D. A. Gregory, T. D. Hudson, “Experiments on nonlinear joint transform correlator using an optically addressed spatial light modulator in the Fourier plane,” Appl. Opt. 30, 1772–1776 (1991).
[CrossRef] [PubMed]

B. Javidi, “Generalization of the linear matched filter concept to nonlinear matched filters,” Appl. Opt. 29, 1215–1224 (1990).
[CrossRef] [PubMed]

B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
[CrossRef] [PubMed]

Laude, V.

Mahalanobis, A.

Painchaud, D.

D. Painchaud, “Distortion invariant nonlinear optical signal processing,” M.S. thesis (University of Connecticut, Storrs, Conn., 1994).

Pratt, W. K.

W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).

Ravichandran, G.

Refregier, P.

Refregier, Ph.

Rotz, F. B.

A. VanderLugt, F. B. Rotz, “The use of film nonlinearities in optical spatial filtering,” Appl. Opt. 1, 215–222 (1970).
[CrossRef]

Tang, Q.

VanderLugt, A.

A. VanderLugt, F. B. Rotz, “The use of film nonlinearities in optical spatial filtering,” Appl. Opt. 1, 215–222 (1970).
[CrossRef]

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

Vijaya Kumar, B. V. K.

Wang, J.

Weaver, C. S.

Willett, P.

Zhang, G.

Appl. Opt. (16)

A. VanderLugt, F. B. Rotz, “The use of film nonlinearities in optical spatial filtering,” Appl. Opt. 1, 215–222 (1970).
[CrossRef]

C. F. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
[CrossRef] [PubMed]

A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
[CrossRef] [PubMed]

B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
[CrossRef] [PubMed]

B. Javidi, “Generalization of the linear matched filter concept to nonlinear matched filters,” Appl. Opt. 29, 1215–1224 (1990).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
[CrossRef]

A. Mahalanobis, D. P. Casasent, “Performance evaluation of minimum average correlation energy filters,” Appl. Opt. 30, 561–572 (1991).
[CrossRef] [PubMed]

B. Javidi, Q. Tang, D. A. Gregory, T. D. Hudson, “Experiments on nonlinear joint transform correlator using an optically addressed spatial light modulator in the Fourier plane,” Appl. Opt. 30, 1772–1776 (1991).
[CrossRef] [PubMed]

D. Casasent, G. Ravichandran, “Advanced distortion-invariant minimum average correlation energy (MACE) filters,” Appl. Opt. 31, 1109–1116 (1992).
[CrossRef] [PubMed]

G. Ravichandran, D. Casasent, “Minimum noise and correlation energy optical correlation filter,” Appl. Opt. 31, 1823–1833 (1992).
[CrossRef] [PubMed]

Q. Tang, B. Javidi, “Sensitivity of the nonlinear joint transform correlator: experimental investigations,” Appl. Opt. 31, 4016–4024 (1992).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, “A tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
[CrossRef]

B. Javidi, J. Wang, A. H. Fazlollahi, “Performance of the nonlinear joint transform correlator for images with low-pass characteristics,” Appl. Opt. 33, 834–848 (1994).
[CrossRef] [PubMed]

B. Javidi, J. Wang, “Limitation of the classic definition of the correlation signal-to-noise ratio in optical pattern recognition with disjoint signal and scene noise,” Appl. Opt. 31, 6826–6829 (1992).
[CrossRef] [PubMed]

J. L. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992).
[CrossRef] [PubMed]

C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
[CrossRef] [PubMed]

IEEE. Trans. Inf. Theory (1)

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

J. Opt. Soc. Am. A (1)

Opt. Comput. Process (1)

Ph. Refregier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with the Wiener approach,” Opt. Comput. Process. 1, 245–266 (1991).

Opt. Eng. (2)

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

Opt. Lett. (3)

Proc. IEEE (1)

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

Other (2)

W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).

D. Painchaud, “Distortion invariant nonlinear optical signal processing,” M.S. thesis (University of Connecticut, Storrs, Conn., 1994).

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Figures (22)

Fig. 1
Fig. 1

(a) Mig reference set member at 0° rotation, (b) Mig reference set member at 45° rotation, (c) F14 input-signal set member at 37.5° rotation, and (d) Mig input-signal set member at 37.5° rotation with noise (σ = 0.2).

Fig. 2
Fig. 2

(a) Lamborghini reference set member at 0° rotation, (b) Lamborghini reference set member at 45° rotation, (c) Ferrari input-signal set member at 37° rotation, and (d) Lamborghini input-signal set member at 33° rotation with noise (σ = 0.2).

Fig. 3
Fig. 3

Mig ECP SDF filter created for k = 1.

Fig. 4
Fig. 4

Mig ECP SDF filter created for k = 0.3.

Fig. 5
Fig. 5

Lamborghini ECP SDF filter created for k = 1.

Fig. 6
Fig. 6

Lamborghini ECP SDF filter created for k = 0.3.

Fig. 7
Fig. 7

JTC output intensity (k = 1) with Mig ECP filter and Mig (37.5°).

Fig. 8
Fig. 8

Same as Fig. 7 except with k = 0.3.

Fig. 9
Fig. 9

JTC output intensity (k = 1) with Lamborghini ECP filter and Lamborghini (33°).

Fig. 10
Fig. 10

Same as Fig. 9 except with k = 0.3.

Fig. 11
Fig. 11

Mig ECP SDF filter performance with noisy input scene (σ = 0.2).

Fig. 12
Fig. 12

Lamborghini ECP SDF filter performance with noisy input scene (σ = 0.2).

Fig. 13
Fig. 13

Mig MACE SDF filter created for k = 1.

Fig. 14
Fig. 14

Mig MACE SDF filter created for k = 0.3.

Fig. 15
Fig. 15

Lamborghini MACE SDF filter created for k = 1.

Fig. 16
Fig. 16

Lamborghini MACE SDF filter created for k = 0.3.

Fig. 17
Fig. 17

JTC output intensity (k = 1) with Mig MACE filter and Mig (37.5°, σ = 0.2).

Fig. 18
Fig. 18

Same as Fig. 17 except with k = 0.3.

Fig. 19
Fig. 19

JTC output intensity (k = 1) with Lamborghini MACE filter and Lamborghini (33°, σ = 0.2).

Fig. 20
Fig. 20

Same as Fig. 19 except with k = 0.3.

Fig. 21
Fig. 21

Mig MACE SDF filter performance with noisy input scene (σ = 0.2).

Fig. 22
Fig. 22

Lamborghini MACE SDF filter performance with noisy input scene (σ = 0.2).

Tables (1)

Tables Icon

Table 1 Summary of Training/Test Image Sets

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

h ECP = S ( S + S ) 1 c * ,
E ave = 1 N i = 1 N n = 1 d | ĥ ( n ) | 2 | ŝ i ( n ) | 2 ,
ĥ MACE = D ̂ 1 Ŝ ( Ŝ + D ̂ 1 Ŝ ) 1 c * ,
D ̂ ( r , r ) = 1 N i = 1 N ŝ i ( r ) ŝ i * ( r ) , r = 1 , , d .
g ( E ) = sgn ( E ) | E | k , k 1 ,
g 1 ( E ) [ | H ( f x , f y ) S ( f x , f y ) | ] k × exp { j [ ϕ H ( f x , f y ) ϕ S ( f x , f y ) ] } ,
ĥ ECP = Ŝ ( Ŝ + Ŝ ) 1 c * ,
v k [ | v [ 1 ] | k exp ( j ϕ v [ 1 ] ) | v [ 2 ] | k exp ( j ϕ v [ 2 ] ) | v [ d ] | k exp ( j ϕ v [ d ] ) ] .
ĥ ECP , k k = Ŝ k [ ( Ŝ k ) + Ŝ k ] 1 c * .
v 1 / k [ | v [ 1 ] | 1 / k exp ( j ϕ v [ 1 ] ) | v [ 2 ] | 1 / k exp ( j ϕ v [ 2 ] ) | v [ d ] | 1 / k exp ( j ϕ v [ d ] ) ] ,
ĥ ECP , k = { Ŝ k [ ( Ŝ k ) + Ŝ k ] 1 c * } 1 / k ,
h ECP , k = F 1 { { Ŝ k [ ( Ŝ k ) + Ŝ k ] 1 c * } 1 / k } .
h MACE , k = F 1 { { ( D ̂ k ) 1 Ŝ k [ ( Ŝ k ) + ( D ̂ k ) 1 Ŝ k ] 1 c * } 1 / k } ,
D ̂ k ( j , k ) = { i = 1 N | ŝ i k ( k ) | 2 j = k = 1 , , d 0 j k .
R 0 2 = max x , y [ | C ( x , y ) | 2 ] ,
SL = max x , y > circular region centered at position of R 0 ( x , y ) and with diameter given by R 0 2 ( x , y ) / 2 [ | C ( x , y ) | 2 ] .
R 0 SL = R 0 2 SL
PCE = [ R 0 2 | C ( x , y ) | 2 ¯ ] { all x , y x , y 0 [ | C ( x , y ) | 2 | C ( x , y ) | 2 ¯ ] 2 ( N x N y 1 ) } 1 / 2 ,
| C ( x , y ) | 2 ¯ = all x , y | C ( x , y ) | 2 N x N y
DR = E { R 0 , target } E { R 0 , nontarget } ,
SNR = E { R 0 2 } E { ( R 0 2 E { R 0 2 } ) 2 } ,
E { R 0 2 } = s R 0 , s 2 N s
E { ( R 0 2 E { R 0 2 } ) 2 } = s ( R 0 , s 2 E { R 0 2 } ) 2 N s

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