Abstract

Previously [ Appl. Opt. 36, p. 9212 (1997)] we examined the performance of the linear and nonlinear preprocessed difference-of-Gaussians filter, and it was shown that this operation results in greater tolerance to in-class variations while maintaining excellent discrimination ability. The introduction of nonlinearity was shown to provide greater robustness to the filter’s response to noise and background clutter in the input scene. We incorporate this new operation into the synthesis of a modified synthetic discriminant function filter. The filter is shown to produce sharp peaks, excellent discrimination without the need to include out-of-class objects, and good invariance to out-of-plane rotation over a distortion range of up to 90°. Additionally, the introduction of nonlinearity is shown to provide greater robustness of the filter response to background clutter in the input scene.

© 1998 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. C. F. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
    [CrossRef] [PubMed]
  3. B. V. K. Vijaya Kumar, “Minimum variance synthetic discriminant functions,” J. Opt. Soc. Am. A 3, 1579–1584 (1986).
    [CrossRef]
  4. A. Mahalanobis, B. V. K. Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef] [PubMed]
  5. D. Casasent, G. Ravichandran, “Advanced distortion-invariant minimum average correlation energy (MACE) filters,” Appl. Opt. 31, 1109–1116 (1992).
    [CrossRef] [PubMed]
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    [CrossRef]
  9. B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. Sims, J. Epperson, “Minimum square error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
    [CrossRef]
  10. A. Mahalanobis, B. V. K. Vijaya Kumar, S. Song, S. R. F. Sims, J. F. Epperson, “Unconstrained correlation filters,” Appl. Opt. 33, 3751–3759 (1994).
    [CrossRef] [PubMed]
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    [CrossRef]
  12. D. Jared, D. Ennis, “Inclusion of filter modulation in the synthetic discriminant function construction,” Appl. Opt. 28, 232–239 (1989).
    [CrossRef] [PubMed]
  13. D. Jared, “Distortion range of filter synthetic discriminant function binary phase-only filters,” Appl. Opt. 28, 4335–4339 (1989).
    [CrossRef]
  14. R. K. Wang, C. R. Chatwin, M. Y. Huang, “Modified filter synthetic discriminant function for improved optical correlator performance,” Appl. Opt. 33, 7646–7654 (1994).
    [CrossRef] [PubMed]
  15. B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
    [CrossRef]
  16. B. V. K. Kumar, A. Mahalanobis, “Recent advances in distortion-invariant correlation filter design,” in Optical Pattern Recognition VI, D. Casasent, T.-H. Chao, eds., Proc. SPIE2490, 2–13 (1995).
    [CrossRef]
  17. D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. Lon. B 207, 187–217 (1980).
    [CrossRef]
  18. L. S. Jamal-Aldin, R. C. D. Young, C. R. Chatwin, “Application of nonlinearity to wavelet-transformed images to improve correlation filter performance,” Appl. Opt. 36, 9212–9224 (1997).
    [CrossRef]
  19. B. Javidi, D. Painchaud, “Distortion-invariant pattern recognition with Fourier-plane nonlinear filters,” Appl. Opt. 35, 318–331 (1996).
    [CrossRef] [PubMed]
  20. A. Mahalanobis, B. V. K. Kumar, “Optimality of the maximum average correlation height filter for detection of targets in noise,” Opt. Eng. 36, 2642–2648 (1997).
    [CrossRef]

1997 (2)

A. Mahalanobis, B. V. K. Kumar, “Optimality of the maximum average correlation height filter for detection of targets in noise,” Opt. Eng. 36, 2642–2648 (1997).
[CrossRef]

L. S. Jamal-Aldin, R. C. D. Young, C. R. Chatwin, “Application of nonlinearity to wavelet-transformed images to improve correlation filter performance,” Appl. Opt. 36, 9212–9224 (1997).
[CrossRef]

1996 (1)

1995 (1)

1994 (2)

1992 (4)

1990 (1)

1989 (2)

D. Jared, D. Ennis, “Inclusion of filter modulation in the synthetic discriminant function construction,” Appl. Opt. 28, 232–239 (1989).
[CrossRef] [PubMed]

D. Jared, “Distortion range of filter synthetic discriminant function binary phase-only filters,” Appl. Opt. 28, 4335–4339 (1989).
[CrossRef]

1987 (1)

1986 (1)

1980 (2)

1964 (1)

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

Brasher, J. D.

B. V. K. Kumar, J. D. Brasher, C. F. Hester, “Nonlinear decision boundaries from the use of complex constraints in synthetic discriminant function filters,” in Photonics for Processors, Neural Networks, and Memories, J. L. Horner, B. Javidi, S. T. Kowel, W. J. Miceli, eds., Proc. SPIE2026, 88–99 (1993).
[CrossRef]

Casasent, D.

Chatwin, C. R.

Ennis, D.

Epperson, J.

B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. Sims, J. Epperson, “Minimum square error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
[CrossRef]

Epperson, J. F.

Flannery, D. L.

Hester, C. F.

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

B. V. K. Kumar, J. D. Brasher, C. F. Hester, “Nonlinear decision boundaries from the use of complex constraints in synthetic discriminant function filters,” in Photonics for Processors, Neural Networks, and Memories, J. L. Horner, B. Javidi, S. T. Kowel, W. J. Miceli, eds., Proc. SPIE2026, 88–99 (1993).
[CrossRef]

Hildreth, E.

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. Lon. B 207, 187–217 (1980).
[CrossRef]

Huang, M. Y.

Jamal-Aldin, L. S.

Jared, D.

D. Jared, “Distortion range of filter synthetic discriminant function binary phase-only filters,” Appl. Opt. 28, 4335–4339 (1989).
[CrossRef]

D. Jared, D. Ennis, “Inclusion of filter modulation in the synthetic discriminant function construction,” Appl. Opt. 28, 232–239 (1989).
[CrossRef] [PubMed]

Javidi, B.

Kumar, B. V. K.

A. Mahalanobis, B. V. K. Kumar, “Optimality of the maximum average correlation height filter for detection of targets in noise,” Opt. Eng. 36, 2642–2648 (1997).
[CrossRef]

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

B. V. K. Kumar, A. Mahalanobis, “Recent advances in distortion-invariant correlation filter design,” in Optical Pattern Recognition VI, D. Casasent, T.-H. Chao, eds., Proc. SPIE2490, 2–13 (1995).
[CrossRef]

B. V. K. Kumar, J. D. Brasher, C. F. Hester, “Nonlinear decision boundaries from the use of complex constraints in synthetic discriminant function filters,” in Photonics for Processors, Neural Networks, and Memories, J. L. Horner, B. Javidi, S. T. Kowel, W. J. Miceli, eds., Proc. SPIE2026, 88–99 (1993).
[CrossRef]

Mahalanobis, A.

A. Mahalanobis, B. V. K. Kumar, “Optimality of the maximum average correlation height filter for detection of targets in noise,” Opt. Eng. 36, 2642–2648 (1997).
[CrossRef]

A. Mahalanobis, B. V. K. Vijaya Kumar, S. Song, S. R. F. Sims, J. F. Epperson, “Unconstrained correlation filters,” Appl. Opt. 33, 3751–3759 (1994).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. Sims, J. Epperson, “Minimum square error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
[CrossRef]

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

B. V. K. Kumar, A. Mahalanobis, “Recent advances in distortion-invariant correlation filter design,” in Optical Pattern Recognition VI, D. Casasent, T.-H. Chao, eds., Proc. SPIE2490, 2–13 (1995).
[CrossRef]

Marr, D.

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. Lon. B 207, 187–217 (1980).
[CrossRef]

Painchaud, D.

Ravichandran, G.

Réfrégier, Ph.

Sims, S.

B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. Sims, J. Epperson, “Minimum square error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
[CrossRef]

Sims, S. R. F.

Song, S.

A. Mahalanobis, B. V. K. Vijaya Kumar, S. Song, S. R. F. Sims, J. F. Epperson, “Unconstrained correlation filters,” Appl. Opt. 33, 3751–3759 (1994).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. Sims, J. Epperson, “Minimum square error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
[CrossRef]

VanderLugt, A.

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

Vijaya Kumar, B. V. K.

Wang, R. K.

Young, R. C. D.

Appl. Opt. (11)

D. Jared, “Distortion range of filter synthetic discriminant function binary phase-only filters,” Appl. Opt. 28, 4335–4339 (1989).
[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. Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
[CrossRef] [PubMed]

D. Jared, D. Ennis, “Inclusion of filter modulation in the synthetic discriminant function construction,” Appl. Opt. 28, 232–239 (1989).
[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 filters,” Appl. Opt. 31, 1823–1833 (1992).
[CrossRef] [PubMed]

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

A. Mahalanobis, B. V. K. Vijaya Kumar, S. Song, S. R. F. Sims, J. F. Epperson, “Unconstrained correlation filters,” Appl. Opt. 33, 3751–3759 (1994).
[CrossRef] [PubMed]

L. S. Jamal-Aldin, R. C. D. Young, C. R. Chatwin, “Application of nonlinearity to wavelet-transformed images to improve correlation filter performance,” Appl. Opt. 36, 9212–9224 (1997).
[CrossRef]

B. Javidi, D. Painchaud, “Distortion-invariant pattern recognition with Fourier-plane nonlinear filters,” Appl. Opt. 35, 318–331 (1996).
[CrossRef] [PubMed]

R. K. Wang, C. R. Chatwin, M. Y. Huang, “Modified filter synthetic discriminant function for improved optical correlator performance,” Appl. Opt. 33, 7646–7654 (1994).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (1)

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

J. Opt. Soc. Am. A (2)

Opt. Eng. (2)

A. Mahalanobis, B. V. K. Kumar, “Optimality of the maximum average correlation height filter for detection of targets in noise,” Opt. Eng. 36, 2642–2648 (1997).
[CrossRef]

B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. Sims, J. Epperson, “Minimum square error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
[CrossRef]

Opt. Lett. (1)

Proc. R. Soc. Lon. B (1)

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. Lon. B 207, 187–217 (1980).
[CrossRef]

Other (2)

B. V. K. Kumar, J. D. Brasher, C. F. Hester, “Nonlinear decision boundaries from the use of complex constraints in synthetic discriminant function filters,” in Photonics for Processors, Neural Networks, and Memories, J. L. Horner, B. Javidi, S. T. Kowel, W. J. Miceli, eds., Proc. SPIE2026, 88–99 (1993).
[CrossRef]

B. V. K. Kumar, A. Mahalanobis, “Recent advances in distortion-invariant correlation filter design,” in Optical Pattern Recognition VI, D. Casasent, T.-H. Chao, eds., Proc. SPIE2490, 2–13 (1995).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Block diagram showing the DOG SDF synthesis.

Fig. 2
Fig. 2

Three views of the test data: An APC rotated out-of-plane by (a) 0°, (b) 45°, and (c) 90°, from an elevation of 45°.

Fig. 3
Fig. 3

Nonlinear sigmoid function.

Fig. 4
Fig. 4

DOG filter used in the computer simulation (a) (σ1, σ2) = (0.8, 0.5) and (b) (σ1, σ2) = (0.5, 0.31).

Fig. 5
Fig. 5

(a) Linear DOG SDF (0.8, 0.5) with the input image at 20°. (b) Linear DOG SDF (0.8, 0.5) with the input image at 18° (intermediate angle). (c) Nonlinear DOG SDF (0.8, 0.5) with the input image at 20°. (d) Nonlinear DOG SDF (0.8, 0.5) with the input image at 18° (intermediate angle). (e) MACH filter.

Fig. 6
Fig. 6

Peak sharpness of the DOG SDF filter.

Fig. 7
Fig. 7

Correlation-peak values obtained with (a) the linear DOG with (σ1, σ2) = (0.8, 0.5), (b) the nonlinear DOG with ξ = 0.10 and (σ1, σ2) = (0.8, 0.5), and (c) the nonlinear DOG with ξ = 1.0 and (σ1, σ2) = (0.8, 0.5).

Fig. 8
Fig. 8

(a) Linear DOG SDF (0.8, 0.5): The input is a training image. (b) Linear DOG SDF (0.8, 0.5): The input is a nontraining in-class (intermediate) image. (c) Nonlinear DOG SDF (0.8, 0.5) with ξ = 0.10: The input is a training image. (d) Nonlinear DOG SDF (0.8, 0.5) with ξ = 0.10: The input is a nontraining in-class (intermediate) image. (e) Nonlinear DOG SDF (0.8, 0.5) with ξ = 1.0: The input is a training image. (f) Nonlinear DOG SDF (0.8, 0.5) with ξ = 1.0: The input is a nontraining in-class (intermediate) image.

Fig. 9
Fig. 9

Correlation-peak values for the two classes, (a) linear and (b) nonlinear, of DOG SDF filters.

Fig. 10
Fig. 10

Input images: Both vehicles are at a rotation angle of 30°. (a) APC. (b) Patton tank.

Fig. 11
Fig. 11

(a) Linear DOG SDF (0.8, 0.5) at 30° rotation. (b) Linear DOG SDF (0.8, 0.5) at 25° rotation. (c) Nonlinear DOG SDF (0.8, 0.5) at 30° rotation. (d) Linear DOG SDF (0.8, 0.5) at 25° rotation.

Fig. 12
Fig. 12

Rotated APC tank embedded in a cluttered scene.

Fig. 13
Fig. 13

Filter response to (a), (c), (e) training and (b), (d), (f) nontraining images embedded in clutter. (a) Linear DOG SDF (0.5, 0.31): The input image with a training image embedded in clutter. (b) Linear DOG SDF (0.5, 0.31): The input image with a nontraining in-class (intermediate) image embedded in clutter (worst case). (c) Nonlinear DOG SDF (0.5, 0.31) with ξ = 0.10: The input image with a training image embedded in clutter. (d) Nonlinear DOG SDF (0.5, 0.31) with ξ = 0.10: The input image with a nontraining in-class (intermediate) image embedded in clutter (worst case). (e) Nonlinear DOG SDF (0.5, 0.31) with ξ = 1.0: The input image with a training image embedded in clutter. (f) Nonlinear DOG SDF (0.5, 0.31) with ξ = 1.0: The input image with a nontraining in-class (intermediate) image embedded in clutter (worst case).

Tables (1)

Tables Icon

Table 1 Target-to-Clutter Ratio (TCR)

Equations (21)

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g i x ,   y = 1 2 π σ i 2 exp - x 2 + y 2 2 π σ i 2 .
g x ,   y = g 1 x ,   y - g 2 x ,   y ,
G u ,   ν = exp - 2 π 2 σ 1 2 u 2 + ν 2 - exp - 2 π 2 σ 2 2 u 2 + ν 2 ,
DOG NL f x ,   y ,   h x ,   y = f x ,   y h x ,   y ,
f x ,   y = NL ( F - 1 F f x ,   y G x ,   y ) ,
T h = C ,
h = a 1 t 1 + a 2 t 2 + + a N t N = T τ a .
T T τ a = C .
a = T T τ - 1 C .
h = T τ T T τ - 1 C .
R = t i x ,   y | t j x ,   y =   t i x ,   y   *   t j x ,   y d x d y , i ,   j = 1 , ,   N ,
h x ,   y = i = 1 N a i t i x ,   y .
N z ;   ξ ,   λ = 1 1 - exp - ξ z - λ .
COPI = max x , y | C x ,   y | 2 ,
PCE = COPI - | C x ,   y | ¯ 2   | C x ,   y | 2 - | C x ,   y | 2 ¯ 2 N x N y - 1 1 / 2 ,
| C x ,   y | 2 ¯ =   | C x ,   y | 2 N x N y
Δ = ACOPI   -   CCOPI ACOPI × 100 % ,
FR = | E C 0 ,   0 TC - E C 0 ,   0 FC | 2 ½ ( var C 0 ,   0 TC + var C 0 ,   0 FC ) ,
TCR = COPI TC HCP ,
COPI TC = max x , y | Y x 1 ,   y 1 | 2 ,
HCP = max x , y | Y x 2 ,   y 2 | 2 ,     x 1 ,   y 1 x 2 ,   y 2 .

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