Abstract

A useful filter for pattern recognition must strike a compromise between the conflicting requirements of in-class distortion tolerance and out-of-class discrimination. Such a filter will be bandpass in nature, the high-frequency response being attenuated to provide less sensitivity to in-class variations, while the low frequencies must be removed, since they compromise the discrimination ability of the filter. A convenient bandpass is the difference of Gaussian (DOG) function, which provides a close approximation to the Laplacian of Gaussian. We describe the effect of a preprocessing operation applied to a DOG filtered image. This operation is shown to result in greater tolerance to in-class variation while maintaining an excellent discrimination ability. Additionally, the introduction of nonlinearity is shown to provide greater robustness in the filter response to noise and background clutter in the input scene.

© 1997 Optical Society of America

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References

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1996

1995

1994

1993

Y. L. Sheng, D. Roberge, H. Szu, T. W. Lu, “Optical wavelet matched filters for shift-invariant pattern recognition,” Opt. Lett. 18, 299–301 (1993).
[CrossRef] [PubMed]

M. W. Wen, S. H. Yin, P. Purwardi, F. T. S. Yu, “Wavelet matched filtering using a photorefractive crystal,” Opt. Commun. 99, 325–330 (1993).
[CrossRef]

R. C. D. Young, C. R. Chatwin, B. F. Scott, “High-speed hybrid optical digital correlator system,” Opt. Eng. 32, 2608–2615 (1993).
[CrossRef]

1992

1990

1989

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

S. Mallat, “A theory of multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 31, 674–693 (1989).
[CrossRef]

1984

A. Grossmann, J. Morlet, “Decomposition of Hardy functions into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723–736 (1984).
[CrossRef]

1982

1980

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

Ahouzi, E.

K. Chalasinska-Macukow, R. Kotynski, K. Styczynski, F. Turon, J. Campos, M. J. Yzuel, E. Ahouzi, C. Gorecki, “Pure phase correlation. Application to optical pattern recognition,” in Euro-American Workshop on Optical Pattern Recognition, B. Javidi, P. Réfrégier, eds. (SPIE Optical Engineering Press, Bellingham, Washington, 1994), pp. 275–292.

Campos, J.

K. Chalasinska-Macukow, R. Kotynski, K. Styczynski, F. Turon, J. Campos, M. J. Yzuel, E. Ahouzi, C. Gorecki, “Pure phase correlation. Application to optical pattern recognition,” in Euro-American Workshop on Optical Pattern Recognition, B. Javidi, P. Réfrégier, eds. (SPIE Optical Engineering Press, Bellingham, Washington, 1994), pp. 275–292.

Caviris, N. P.

Chalasinska-Macukow, K.

K. Chalasinska-Macukow, R. Kotynski, K. Styczynski, F. Turon, J. Campos, M. J. Yzuel, E. Ahouzi, C. Gorecki, “Pure phase correlation. Application to optical pattern recognition,” in Euro-American Workshop on Optical Pattern Recognition, B. Javidi, P. Réfrégier, eds. (SPIE Optical Engineering Press, Bellingham, Washington, 1994), pp. 275–292.

Chatwin, C. R.

R. C. D. Young, C. R. Chatwin, B. F. Scott, “High-speed hybrid optical digital correlator system,” Opt. Eng. 32, 2608–2615 (1993).
[CrossRef]

R. C. D. Young, C. R. Chatwin, “Design and simulation of a synthetic discriminant function filter for implementation in an up-dateable photorefractive correlator,” in Optical Pattern Recognition III, D. Casasent, T.-H. Chao, eds., Proc. SPIE1701, 239–263 (1992).
[CrossRef]

Chui, C. K.

C. K. Chui, An Introduction to Wavelets (Academic, San Diego, Calif., 1992).

Daubechies, I.

I. Daubechies, “The wavelet transform, time frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

I. Daubechies, Ten Lectures on Wavelets (Academic, San Diego, Calif., 1992).
[CrossRef]

Epperson, J.

Flannery, D.

Gorecki, C.

K. Chalasinska-Macukow, R. Kotynski, K. Styczynski, F. Turon, J. Campos, M. J. Yzuel, E. Ahouzi, C. Gorecki, “Pure phase correlation. Application to optical pattern recognition,” in Euro-American Workshop on Optical Pattern Recognition, B. Javidi, P. Réfrégier, eds. (SPIE Optical Engineering Press, Bellingham, Washington, 1994), pp. 275–292.

Grossmann, A.

A. Grossmann, J. Morlet, “Decomposition of Hardy functions into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723–736 (1984).
[CrossRef]

Hassebrook, L.

Hildreth, E.

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

Horner, J. L.

Javidi, B.

Kanterakis, E. G.

Katz, A.

Kotynski, R.

K. Chalasinska-Macukow, R. Kotynski, K. Styczynski, F. Turon, J. Campos, M. J. Yzuel, E. Ahouzi, C. Gorecki, “Pure phase correlation. Application to optical pattern recognition,” in Euro-American Workshop on Optical Pattern Recognition, B. Javidi, P. Réfrégier, eds. (SPIE Optical Engineering Press, Bellingham, Washington, 1994), pp. 275–292.

Kumar, B. V. K.

Lu, T. W.

Lu, X. J.

Mahalanobis, A.

Mallat, S.

S. Mallat, “A theory of multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 31, 674–693 (1989).
[CrossRef]

Marr, D.

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

Meyer, Y.

Y. Meyer, Waveletsand Operators (Cambridge U. Press, Cambridge, UK, 1992).

Morlet, J.

A. Grossmann, J. Morlet, “Decomposition of Hardy functions into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723–736 (1984).
[CrossRef]

Painchaud, D.

Purwardi, P.

M. W. Wen, S. H. Yin, P. Purwardi, F. T. S. Yu, “Wavelet matched filtering using a photorefractive crystal,” Opt. Commun. 99, 325–330 (1993).
[CrossRef]

Refregier, P.

Roberge, D.

Scott, B. F.

R. C. D. Young, C. R. Chatwin, B. F. Scott, “High-speed hybrid optical digital correlator system,” Opt. Eng. 32, 2608–2615 (1993).
[CrossRef]

Sheng, Y. L.

Sims, S. R. F.

Song, S.

Styczynski, K.

K. Chalasinska-Macukow, R. Kotynski, K. Styczynski, F. Turon, J. Campos, M. J. Yzuel, E. Ahouzi, C. Gorecki, “Pure phase correlation. Application to optical pattern recognition,” in Euro-American Workshop on Optical Pattern Recognition, B. Javidi, P. Réfrégier, eds. (SPIE Optical Engineering Press, Bellingham, Washington, 1994), pp. 275–292.

Szu, H.

Turon, F.

K. Chalasinska-Macukow, R. Kotynski, K. Styczynski, F. Turon, J. Campos, M. J. Yzuel, E. Ahouzi, C. Gorecki, “Pure phase correlation. Application to optical pattern recognition,” in Euro-American Workshop on Optical Pattern Recognition, B. Javidi, P. Réfrégier, eds. (SPIE Optical Engineering Press, Bellingham, Washington, 1994), pp. 275–292.

Vijaya Kumar, B. V. K.

Wen, M. W.

M. W. Wen, S. H. Yin, P. Purwardi, F. T. S. Yu, “Wavelet matched filtering using a photorefractive crystal,” Opt. Commun. 99, 325–330 (1993).
[CrossRef]

Yin, S. H.

M. W. Wen, S. H. Yin, P. Purwardi, F. T. S. Yu, “Wavelet matched filtering using a photorefractive crystal,” Opt. Commun. 99, 325–330 (1993).
[CrossRef]

Young, R. C. D.

R. C. D. Young, C. R. Chatwin, B. F. Scott, “High-speed hybrid optical digital correlator system,” Opt. Eng. 32, 2608–2615 (1993).
[CrossRef]

R. C. D. Young, C. R. Chatwin, “Design and simulation of a synthetic discriminant function filter for implementation in an up-dateable photorefractive correlator,” in Optical Pattern Recognition III, D. Casasent, T.-H. Chao, eds., Proc. SPIE1701, 239–263 (1992).
[CrossRef]

Yu, F. T. S.

M. W. Wen, S. H. Yin, P. Purwardi, F. T. S. Yu, “Wavelet matched filtering using a photorefractive crystal,” Opt. Commun. 99, 325–330 (1993).
[CrossRef]

Yzuel, M. J.

K. Chalasinska-Macukow, R. Kotynski, K. Styczynski, F. Turon, J. Campos, M. J. Yzuel, E. Ahouzi, C. Gorecki, “Pure phase correlation. Application to optical pattern recognition,” in Euro-American Workshop on Optical Pattern Recognition, B. Javidi, P. Réfrégier, eds. (SPIE Optical Engineering Press, Bellingham, Washington, 1994), pp. 275–292.

Zhang, G.

B. Javidi, G. Zhang, “Experiments on non-linearly transformed matched filters,” Opt. Eng. 31, 934–938 (1992).
[CrossRef]

Appl. Opt.

IEEE Trans. Inf. Theory

I. Daubechies, “The wavelet transform, time frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell.

S. Mallat, “A theory of multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 31, 674–693 (1989).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

M. W. Wen, S. H. Yin, P. Purwardi, F. T. S. Yu, “Wavelet matched filtering using a photorefractive crystal,” Opt. Commun. 99, 325–330 (1993).
[CrossRef]

Opt. Eng.

R. C. D. Young, C. R. Chatwin, B. F. Scott, “High-speed hybrid optical digital correlator system,” Opt. Eng. 32, 2608–2615 (1993).
[CrossRef]

B. Javidi, G. Zhang, “Experiments on non-linearly transformed matched filters,” Opt. Eng. 31, 934–938 (1992).
[CrossRef]

Opt. Lett.

Proc. RT. Soc. London Ser. B

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

SIAM J. Math. Anal.

A. Grossmann, J. Morlet, “Decomposition of Hardy functions into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723–736 (1984).
[CrossRef]

Other

R. C. D. Young, C. R. Chatwin, “Design and simulation of a synthetic discriminant function filter for implementation in an up-dateable photorefractive correlator,” in Optical Pattern Recognition III, D. Casasent, T.-H. Chao, eds., Proc. SPIE1701, 239–263 (1992).
[CrossRef]

I. Daubechies, Ten Lectures on Wavelets (Academic, San Diego, Calif., 1992).
[CrossRef]

C. K. Chui, An Introduction to Wavelets (Academic, San Diego, Calif., 1992).

Y. Meyer, Waveletsand Operators (Cambridge U. Press, Cambridge, UK, 1992).

K. Chalasinska-Macukow, R. Kotynski, K. Styczynski, F. Turon, J. Campos, M. J. Yzuel, E. Ahouzi, C. Gorecki, “Pure phase correlation. Application to optical pattern recognition,” in Euro-American Workshop on Optical Pattern Recognition, B. Javidi, P. Réfrégier, eds. (SPIE Optical Engineering Press, Bellingham, Washington, 1994), pp. 275–292.

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Figures (18)

Fig. 1
Fig. 1

Cross section of the DOG wavelet in the (a) space domain and (b) frequency domain.

Fig. 2
Fig. 2

(a) Image showing the concentration of energy at enclosed areas of the image. (b) Intensity variation of the same image in isometric form.

Fig. 3
Fig. 3

Block diagram showing the operations required to implement a correlation using the modified filter function.

Fig. 4
Fig. 4

Nonlinearity sigmoid function used.

Fig. 5
Fig. 5

DOG functions for selected parameters.

Fig. 6
Fig. 6

Autocorrelation peaks for (a) a POF, (b) a CMF, (c) a linear DOG, and (d) a nonlinear DOG.

Fig. 7
Fig. 7

APC (a) reference image and (b) input image with the APC rotated by 20°.

Fig. 8
Fig. 8

Effect of in-plane rotation on the correlation peak height.

Fig. 9
Fig. 9

Effect of nonlinearity on the distortion invariance of the modified DOG.

Fig. 10
Fig. 10

APC at an elevation of 45°: (a) reference image and (b) input image with the APC rotated by 20°. APC at an elevation of 60°: (c) reference image and (d) input image with the APC rotated by 20°.

Fig. 11
Fig. 11

Effect of out-of-plane rotation (45°) on correlation peak height.

Fig. 12
Fig. 12

Effect of out-of-plane rotation (60°) on correlation peak height.

Fig. 13
Fig. 13

(a) Target and (b) nontarget images used to test discrimination ability.

Fig. 14
Fig. 14

Discrimination ability of (a) a POF, (b) a CMF, (c) a linear DOG (0.5, 0.31), and (d) a nonlinear DOG (0.5, 0.31) ξ = 0.15.

Fig. 15
Fig. 15

Horner efficiency for the different filters.

Fig. 16
Fig. 16

(a) Noise-free image of the APC vehicle. (b) Noise-corrupted image of the APC vehicle with a noise variance of σ = 0.02.

Fig. 17
Fig. 17

APC vehicle embedded in background clutter noise.

Fig. 18
Fig. 18

(a) Performance of the linear DOG filter in background clutter. (b) Performance of the nonlinear DOG filter in background clutter.

Tables (7)

Tables Icon

Table 1 COPI and PCE Values of Autocorrelation for the Different Filters

Tables Icon

Table 2 Effect of In-Plane Rotation on the Correlation Peak Height and the PCE Value

Tables Icon

Table 3 Effect of Out-of-Plane Rotation on the Correlation Peak Height

Tables Icon

Table 4 Discrimination Ability for the Different Filters

Tables Icon

Table 5 Effect of Varying the Bandpass of the DOG Filter on the Horner Efficiency

Tables Icon

Table 6 Noise Tolerance Ability of the Different Filters

Tables Icon

Table 7 Qualitative Summary of the Comparative Filter Performance

Equations (15)

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

Wgfs,τ1s1/2-ftg*t-τsdt.
gix, y=12πσi2exp-x2+y22πσi2.
gx, y=g1x, y-g2x, y,
Gu, v=exp-2π2σ12u2+v2-exp-2π2σ22u2+v2,
DOGNLfx, y, hx, y=ηfx, yηhx, y,
ηfx, y=NLF-1Ffx,yGu,v,
Nz; ξ, λ=11-exp-ξz-λ.
COPI=maxx,yCx, y2,
PCE=COPIEnergyc,
Energyc=-Cx, y2dx.
Δ=ACOPI-CCOPIACOPI×100%,
ηH=COPIIx, y2,
SNR=ECOPIECOPI-ECOPI21/2,
ECOPI=sCOPIsNs
ECOPI-ECOPI2=sCOPIs-ECOPI2Ns

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