Abstract

We present a statistical pattern recognition scheme for detecting vehicles in still images. The methodology involves pattern classification using higher-order statistics (HOS) in a clustering framework. The proposed method approximately models the unknown distribution of the image patterns of vehicles by learning HOS information about the vehicle class from sample images. Given a test image, statistical information about the background is learned “on the fly.” An HOS-based decision measure derived from a series expansion of the multivariate probability density function in terms of the Gaussian function and Hermite polynomials is used to classify test patterns as vehicles or otherwise. Experimental results on real images with cluttered background are given to demonstrate the performance of the proposed method. When tested on real aerial images, the method gives good results, even for complicated scenes. The detection rate is found to be quite good, while the false alarms are very few. The method can serve as an important step toward building an automated traffic monitoring system.

© 2001 Optical Society of America

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References

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  1. R. Chellappa, Q. Zheng, L. Davis, C. Lin, X. Zhang, C. Rodriguez, A. Rosenfeld, T. Moore, “Site model based monitoring of aerial images,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1994), pp. 295–318.
  2. P. Burlina, V. Parameswaran, R. Chellappa, “Sensitivity analysis and learning strategies for context-based vehicle detection algorithms,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1997), pp. 577–583.
  3. P. Burlina, R. Chellappa, C. L. Lin, “A spectral attentional mechanism tuned to object configurations,” IEEE Trans. Image Process. 6, 117–1128 (1997).
    [CrossRef]
  4. H. Moon, R. Chellappa, A. Rosenfeld, “Performance analysis of a simple vehicle detection algorithm,” in Proceedings of the FedLab Symposium on Advanced Sensors (U.S. Army Research Laboratory, Adelphi, Md., 1999), pp. 249–253.
  5. D. H. Huttenlocher, R. Zabih, “Aerial and ground-based video surveillance at Cornell University,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos Calif., 1998), pp. 77–83.
  6. L. S. Davis, R. Chellappa, A. Rosenfeld, D. Harwood, I. Haritaoglu, R. Cutler, “Visual surveillance and monitoring,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 73–76.
  7. T. Kanade, R. T. Collins, A. J. Lipton, P. Burt, L. Wixson, “Advances in cooperative multi-sensor video surveillance,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 3–24.
  8. A. J. Lipton, H. Fujiyoshi, R. S. Patil, “Moving target classification and tracking from real-time video,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 129–136.
  9. R. Cutler, L. Davis, “Real time periodic motion detection, analysis and applications,” in Proceedings on Computer Vision and Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1999), pp. 326–332.
  10. A. N. Rajagopalan, R. Chellappa, “Higher-order spectral analysis of human motion,” in Proceedings of the International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2000), Vol. 3, pp. 230–233.
  11. A. N. Rajagopalan, P. Burlina, R. Chellappa, “Higher-order statistical learning for vehicle detection in images,” in Proceedings of the IEEE International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 1204–1209.
  12. R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).
  13. T. Y. Young, T. W. Calvert, Classification, Estimation and Pattern Recognition (American Elsevier, New York, 1974).
  14. C. H. Chen, “Theoretical comparison of a class of feature selection criteria in pattern recognition,” IEEE Trans. Comput. C-20, 1054–1056 (1971).
    [CrossRef]
  15. M. E. Hellman, “Probability of error, equivocation, and Chernoff bound,” IEEE Trans. Inf. Theory IT-16, 368–372 (1970).
    [CrossRef]
  16. P. A. Devijver, “On a new class of bounds on Bayes risk in multi-hypothesis pattern recognition,” IEEE Trans. Comput. C-23, 70–80 (1974).
    [CrossRef]
  17. D. G. Lainiotis, S. K. Park, “Probability of error bounds,” IEEE Trans. Syst. Man Cybern. SMC-1, 175–178 (1971).
  18. K. Sung, T. Poggio, “Example-based learning for view-based human face detection,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 39–51 (1998).
    [CrossRef]
  19. A. K. Jain, R. C. Dubes, Algorithms for Clustering Data (Prentice-Hall, Englewood Cliffs, N. J., 1988).
  20. G. M. Kendall, A. Stuart, The Advanced Theory of Statistics (Charles Griffin, London, 1958), Vol. 1.

1998 (1)

K. Sung, T. Poggio, “Example-based learning for view-based human face detection,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 39–51 (1998).
[CrossRef]

1997 (1)

P. Burlina, R. Chellappa, C. L. Lin, “A spectral attentional mechanism tuned to object configurations,” IEEE Trans. Image Process. 6, 117–1128 (1997).
[CrossRef]

1974 (1)

P. A. Devijver, “On a new class of bounds on Bayes risk in multi-hypothesis pattern recognition,” IEEE Trans. Comput. C-23, 70–80 (1974).
[CrossRef]

1971 (2)

D. G. Lainiotis, S. K. Park, “Probability of error bounds,” IEEE Trans. Syst. Man Cybern. SMC-1, 175–178 (1971).

C. H. Chen, “Theoretical comparison of a class of feature selection criteria in pattern recognition,” IEEE Trans. Comput. C-20, 1054–1056 (1971).
[CrossRef]

1970 (1)

M. E. Hellman, “Probability of error, equivocation, and Chernoff bound,” IEEE Trans. Inf. Theory IT-16, 368–372 (1970).
[CrossRef]

Burlina, P.

P. Burlina, R. Chellappa, C. L. Lin, “A spectral attentional mechanism tuned to object configurations,” IEEE Trans. Image Process. 6, 117–1128 (1997).
[CrossRef]

P. Burlina, V. Parameswaran, R. Chellappa, “Sensitivity analysis and learning strategies for context-based vehicle detection algorithms,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1997), pp. 577–583.

A. N. Rajagopalan, P. Burlina, R. Chellappa, “Higher-order statistical learning for vehicle detection in images,” in Proceedings of the IEEE International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 1204–1209.

Burt, P.

T. Kanade, R. T. Collins, A. J. Lipton, P. Burt, L. Wixson, “Advances in cooperative multi-sensor video surveillance,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 3–24.

Calvert, T. W.

T. Y. Young, T. W. Calvert, Classification, Estimation and Pattern Recognition (American Elsevier, New York, 1974).

Chellappa, R.

P. Burlina, R. Chellappa, C. L. Lin, “A spectral attentional mechanism tuned to object configurations,” IEEE Trans. Image Process. 6, 117–1128 (1997).
[CrossRef]

L. S. Davis, R. Chellappa, A. Rosenfeld, D. Harwood, I. Haritaoglu, R. Cutler, “Visual surveillance and monitoring,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 73–76.

H. Moon, R. Chellappa, A. Rosenfeld, “Performance analysis of a simple vehicle detection algorithm,” in Proceedings of the FedLab Symposium on Advanced Sensors (U.S. Army Research Laboratory, Adelphi, Md., 1999), pp. 249–253.

A. N. Rajagopalan, R. Chellappa, “Higher-order spectral analysis of human motion,” in Proceedings of the International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2000), Vol. 3, pp. 230–233.

A. N. Rajagopalan, P. Burlina, R. Chellappa, “Higher-order statistical learning for vehicle detection in images,” in Proceedings of the IEEE International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 1204–1209.

P. Burlina, V. Parameswaran, R. Chellappa, “Sensitivity analysis and learning strategies for context-based vehicle detection algorithms,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1997), pp. 577–583.

R. Chellappa, Q. Zheng, L. Davis, C. Lin, X. Zhang, C. Rodriguez, A. Rosenfeld, T. Moore, “Site model based monitoring of aerial images,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1994), pp. 295–318.

Chen, C. H.

C. H. Chen, “Theoretical comparison of a class of feature selection criteria in pattern recognition,” IEEE Trans. Comput. C-20, 1054–1056 (1971).
[CrossRef]

Collins, R. T.

T. Kanade, R. T. Collins, A. J. Lipton, P. Burt, L. Wixson, “Advances in cooperative multi-sensor video surveillance,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 3–24.

Cutler, R.

L. S. Davis, R. Chellappa, A. Rosenfeld, D. Harwood, I. Haritaoglu, R. Cutler, “Visual surveillance and monitoring,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 73–76.

R. Cutler, L. Davis, “Real time periodic motion detection, analysis and applications,” in Proceedings on Computer Vision and Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1999), pp. 326–332.

Davis, L.

R. Cutler, L. Davis, “Real time periodic motion detection, analysis and applications,” in Proceedings on Computer Vision and Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1999), pp. 326–332.

R. Chellappa, Q. Zheng, L. Davis, C. Lin, X. Zhang, C. Rodriguez, A. Rosenfeld, T. Moore, “Site model based monitoring of aerial images,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1994), pp. 295–318.

Davis, L. S.

L. S. Davis, R. Chellappa, A. Rosenfeld, D. Harwood, I. Haritaoglu, R. Cutler, “Visual surveillance and monitoring,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 73–76.

Devijver, P. A.

P. A. Devijver, “On a new class of bounds on Bayes risk in multi-hypothesis pattern recognition,” IEEE Trans. Comput. C-23, 70–80 (1974).
[CrossRef]

Dubes, R. C.

A. K. Jain, R. C. Dubes, Algorithms for Clustering Data (Prentice-Hall, Englewood Cliffs, N. J., 1988).

Duda, R. O.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Fujiyoshi, H.

A. J. Lipton, H. Fujiyoshi, R. S. Patil, “Moving target classification and tracking from real-time video,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 129–136.

Haritaoglu, I.

L. S. Davis, R. Chellappa, A. Rosenfeld, D. Harwood, I. Haritaoglu, R. Cutler, “Visual surveillance and monitoring,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 73–76.

Hart, P. E.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Harwood, D.

L. S. Davis, R. Chellappa, A. Rosenfeld, D. Harwood, I. Haritaoglu, R. Cutler, “Visual surveillance and monitoring,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 73–76.

Hellman, M. E.

M. E. Hellman, “Probability of error, equivocation, and Chernoff bound,” IEEE Trans. Inf. Theory IT-16, 368–372 (1970).
[CrossRef]

Huttenlocher, D. H.

D. H. Huttenlocher, R. Zabih, “Aerial and ground-based video surveillance at Cornell University,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos Calif., 1998), pp. 77–83.

Jain, A. K.

A. K. Jain, R. C. Dubes, Algorithms for Clustering Data (Prentice-Hall, Englewood Cliffs, N. J., 1988).

Kanade, T.

T. Kanade, R. T. Collins, A. J. Lipton, P. Burt, L. Wixson, “Advances in cooperative multi-sensor video surveillance,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 3–24.

Kendall, G. M.

G. M. Kendall, A. Stuart, The Advanced Theory of Statistics (Charles Griffin, London, 1958), Vol. 1.

Lainiotis, D. G.

D. G. Lainiotis, S. K. Park, “Probability of error bounds,” IEEE Trans. Syst. Man Cybern. SMC-1, 175–178 (1971).

Lin, C.

R. Chellappa, Q. Zheng, L. Davis, C. Lin, X. Zhang, C. Rodriguez, A. Rosenfeld, T. Moore, “Site model based monitoring of aerial images,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1994), pp. 295–318.

Lin, C. L.

P. Burlina, R. Chellappa, C. L. Lin, “A spectral attentional mechanism tuned to object configurations,” IEEE Trans. Image Process. 6, 117–1128 (1997).
[CrossRef]

Lipton, A. J.

T. Kanade, R. T. Collins, A. J. Lipton, P. Burt, L. Wixson, “Advances in cooperative multi-sensor video surveillance,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 3–24.

A. J. Lipton, H. Fujiyoshi, R. S. Patil, “Moving target classification and tracking from real-time video,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 129–136.

Moon, H.

H. Moon, R. Chellappa, A. Rosenfeld, “Performance analysis of a simple vehicle detection algorithm,” in Proceedings of the FedLab Symposium on Advanced Sensors (U.S. Army Research Laboratory, Adelphi, Md., 1999), pp. 249–253.

Moore, T.

R. Chellappa, Q. Zheng, L. Davis, C. Lin, X. Zhang, C. Rodriguez, A. Rosenfeld, T. Moore, “Site model based monitoring of aerial images,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1994), pp. 295–318.

Parameswaran, V.

P. Burlina, V. Parameswaran, R. Chellappa, “Sensitivity analysis and learning strategies for context-based vehicle detection algorithms,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1997), pp. 577–583.

Park, S. K.

D. G. Lainiotis, S. K. Park, “Probability of error bounds,” IEEE Trans. Syst. Man Cybern. SMC-1, 175–178 (1971).

Patil, R. S.

A. J. Lipton, H. Fujiyoshi, R. S. Patil, “Moving target classification and tracking from real-time video,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 129–136.

Poggio, T.

K. Sung, T. Poggio, “Example-based learning for view-based human face detection,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 39–51 (1998).
[CrossRef]

Rajagopalan, A. N.

A. N. Rajagopalan, R. Chellappa, “Higher-order spectral analysis of human motion,” in Proceedings of the International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2000), Vol. 3, pp. 230–233.

A. N. Rajagopalan, P. Burlina, R. Chellappa, “Higher-order statistical learning for vehicle detection in images,” in Proceedings of the IEEE International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 1204–1209.

Rodriguez, C.

R. Chellappa, Q. Zheng, L. Davis, C. Lin, X. Zhang, C. Rodriguez, A. Rosenfeld, T. Moore, “Site model based monitoring of aerial images,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1994), pp. 295–318.

Rosenfeld, A.

R. Chellappa, Q. Zheng, L. Davis, C. Lin, X. Zhang, C. Rodriguez, A. Rosenfeld, T. Moore, “Site model based monitoring of aerial images,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1994), pp. 295–318.

H. Moon, R. Chellappa, A. Rosenfeld, “Performance analysis of a simple vehicle detection algorithm,” in Proceedings of the FedLab Symposium on Advanced Sensors (U.S. Army Research Laboratory, Adelphi, Md., 1999), pp. 249–253.

L. S. Davis, R. Chellappa, A. Rosenfeld, D. Harwood, I. Haritaoglu, R. Cutler, “Visual surveillance and monitoring,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 73–76.

Stuart, A.

G. M. Kendall, A. Stuart, The Advanced Theory of Statistics (Charles Griffin, London, 1958), Vol. 1.

Sung, K.

K. Sung, T. Poggio, “Example-based learning for view-based human face detection,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 39–51 (1998).
[CrossRef]

Wixson, L.

T. Kanade, R. T. Collins, A. J. Lipton, P. Burt, L. Wixson, “Advances in cooperative multi-sensor video surveillance,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 3–24.

Young, T. Y.

T. Y. Young, T. W. Calvert, Classification, Estimation and Pattern Recognition (American Elsevier, New York, 1974).

Zabih, R.

D. H. Huttenlocher, R. Zabih, “Aerial and ground-based video surveillance at Cornell University,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos Calif., 1998), pp. 77–83.

Zhang, X.

R. Chellappa, Q. Zheng, L. Davis, C. Lin, X. Zhang, C. Rodriguez, A. Rosenfeld, T. Moore, “Site model based monitoring of aerial images,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1994), pp. 295–318.

Zheng, Q.

R. Chellappa, Q. Zheng, L. Davis, C. Lin, X. Zhang, C. Rodriguez, A. Rosenfeld, T. Moore, “Site model based monitoring of aerial images,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1994), pp. 295–318.

IEEE Trans. Comput. (2)

C. H. Chen, “Theoretical comparison of a class of feature selection criteria in pattern recognition,” IEEE Trans. Comput. C-20, 1054–1056 (1971).
[CrossRef]

P. A. Devijver, “On a new class of bounds on Bayes risk in multi-hypothesis pattern recognition,” IEEE Trans. Comput. C-23, 70–80 (1974).
[CrossRef]

IEEE Trans. Image Process. (1)

P. Burlina, R. Chellappa, C. L. Lin, “A spectral attentional mechanism tuned to object configurations,” IEEE Trans. Image Process. 6, 117–1128 (1997).
[CrossRef]

IEEE Trans. Inf. Theory (1)

M. E. Hellman, “Probability of error, equivocation, and Chernoff bound,” IEEE Trans. Inf. Theory IT-16, 368–372 (1970).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

K. Sung, T. Poggio, “Example-based learning for view-based human face detection,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 39–51 (1998).
[CrossRef]

IEEE Trans. Syst. Man Cybern. (1)

D. G. Lainiotis, S. K. Park, “Probability of error bounds,” IEEE Trans. Syst. Man Cybern. SMC-1, 175–178 (1971).

Other (14)

A. K. Jain, R. C. Dubes, Algorithms for Clustering Data (Prentice-Hall, Englewood Cliffs, N. J., 1988).

G. M. Kendall, A. Stuart, The Advanced Theory of Statistics (Charles Griffin, London, 1958), Vol. 1.

R. Chellappa, Q. Zheng, L. Davis, C. Lin, X. Zhang, C. Rodriguez, A. Rosenfeld, T. Moore, “Site model based monitoring of aerial images,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1994), pp. 295–318.

P. Burlina, V. Parameswaran, R. Chellappa, “Sensitivity analysis and learning strategies for context-based vehicle detection algorithms,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1997), pp. 577–583.

H. Moon, R. Chellappa, A. Rosenfeld, “Performance analysis of a simple vehicle detection algorithm,” in Proceedings of the FedLab Symposium on Advanced Sensors (U.S. Army Research Laboratory, Adelphi, Md., 1999), pp. 249–253.

D. H. Huttenlocher, R. Zabih, “Aerial and ground-based video surveillance at Cornell University,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos Calif., 1998), pp. 77–83.

L. S. Davis, R. Chellappa, A. Rosenfeld, D. Harwood, I. Haritaoglu, R. Cutler, “Visual surveillance and monitoring,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 73–76.

T. Kanade, R. T. Collins, A. J. Lipton, P. Burt, L. Wixson, “Advances in cooperative multi-sensor video surveillance,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 3–24.

A. J. Lipton, H. Fujiyoshi, R. S. Patil, “Moving target classification and tracking from real-time video,” in Proceedings of the DARPA Image Understanding Workshop (Morgan Kaufmann, Los Altos, Calif., 1998), pp. 129–136.

R. Cutler, L. Davis, “Real time periodic motion detection, analysis and applications,” in Proceedings on Computer Vision and Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1999), pp. 326–332.

A. N. Rajagopalan, R. Chellappa, “Higher-order spectral analysis of human motion,” in Proceedings of the International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2000), Vol. 3, pp. 230–233.

A. N. Rajagopalan, P. Burlina, R. Chellappa, “Higher-order statistical learning for vehicle detection in images,” in Proceedings of the IEEE International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 1204–1209.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

T. Y. Young, T. W. Calvert, Classification, Estimation and Pattern Recognition (American Elsevier, New York, 1974).

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Figures (5)

Fig. 1
Fig. 1

Some training examples for learning the vehicle class.

Fig. 2
Fig. 2

Background patterns obtained on the fly by using the dynamic background learning algorithm. Note that none of these image patterns is of vehicles.

Fig. 3
Fig. 3

Vehicle detection results using Mahalanobis distance without background learning. Multiple boxes represent detections at different scales. Below each image, three values are given; they correspond to the number of vehicles detected, the number of vehicles present in the image, and the number of false alarms, respectively. Vehicles that are much smaller than the training image are ignored in the calculations.

Fig. 4
Fig. 4

Results of the HOS-based vehicle detection scheme without background learning. Below each image, three values are given; they correspond to the number of vehicles detected, the number of vehicles present in the image, and the number of false alarms, respectively. Vehicles that are much smaller than the training image are ignored in the calculations.

Fig. 5
Fig. 5

Same as Fig. 4 but with dynamic background learning.

Tables (1)

Tables Icon

Table 1 Performance Comparison

Equations (63)

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

H1:x belongstovehicleclassω1,
H2:x belongstobackgroundclassω2.
Iffx(x|ω1)P(ω1)>fx(x|ω2)P(ω2)then xω1,
Iffx(x|ω2)P(ω2)>fx(x|ω1)P(ω1),then xω2,
Pe=P(xR1, ω2)+P(xR2, ω1)=P(xR1|ω2)P(ω2)+P(xR2|ω1)P(ω1)=pR2fx|ω1 dx(1-p)R1fx|ω2 dx,
Pe= min[pf1, (1-p)f2]dx,
min(α1, α2)α1α2.
Pe= min[pf1,(1-p)f2]dx[p(1-p)]1/2[f1(x)f2(x)]1/2 dx.
Pe12[f1(x)f2(x)]1/2 dx.
Φ(t)=E[exp(tTX)].
E[exp(tTX-12tTt)exp(sTX-12sTs)]=exp(tTs).
g(x)=n=01n!(xn)T[Dxng(x)]x=0,
exptTx-12tTt=n=01n!(tn)TH¯n(x),
Em,n=0(tn)TH¯n(X)n!H¯mT(X)m!sm=n=01n!(tn)Tsn=m,n=01n!(tn)TIqnqmsm,  Iqnqm=OqnqmfornmIqnforn=m.
E[Hn(X)HmT(X)]=Ipnpm.
H3(x1, x2)=H3(x1)3!H2(x1)2!H1(x2)1!×H1(x1)1!H2(x2)2!H3(x2)3!T,
H2(x1, x2)=H2(x1)2!H1(x1)1!H1(x2)1!H2(x2)2!T,
1(2π)N/2exp-12xTxHn(x)HmT(x)dx=Ipnpm.
1(2π)N/2(det R)1/2exp-12(y-μ)TR-1(y-μ)×Hn(R-1/2(y-μ))HmT(R-1/2(y-μ))dy=Ipnpm.
E[Hn(R-1/2(X-μ))HmT(R-1/2(X-μ))]=Ipnpm,
f(x)=N(0, I)m=0CmHm(x).
f(x)HnT(x)dx=m=0CmN(0, I)Hm(x)HnT(x)dx
=CnIn[from Eq.(6)].
Cn=f(x)HnT(x)dx=E[HnT(X)].
f(x)=N(0, I)1+n=1E[HnT(X)]Hn(x).
f(x)=N(μ, R)1+n=1E[HnT(R-1/2(X-μ))]×Hn(R-1/2(x-μ)).
f(x)=N(μ, R)1+n=3E[HnT(R-1/2(X-μ))]×Hn(R-1/2(x-μ)).
12[f1(x)f2(x)]1/2 dx.
β=[f1(x)f2(x)]1/2 dx.
0β1,
fi(x)=1(2π)N/2(det Ri)1/2exp-12(x-μi)TRi-1(x-μi)×[1+Si(x)],i=1, 2,
12QE{[1+S1(x)][1+S2(x)]}1/2,
Si(x)=m=3Ei{[Hm(Ri-1/2(X-μi)T)×Hm(Ri-1/2(x-μi))]}
Q=exp-14(μ1-μ2)T(R1+R2)-1(μ1-μ2)×(det R1R2)1/4detR1+R221/2.
β=1(2π)N/2(det R1)1/4(det R2)1/4×exp-14[(x-μ1)TR1-1(x-μ1)+(x-μ2)TR2-1(x-μ2)]×{[1+S1(x)][1+S1(x)]}1/2 dx.
A=(x-μ1)TR1-1(x-μ1)+(x-μ2)TR2-1(x-μ2)=xT(R1-1+R2-1)x-2xT(R1-1μ1+R2-1μ2)+μ1TR1-1μ1+μ2TR2-1μ2.
μ=(R1-1+R2-1)-1(R1-1μ1+R2-1μ2),
μT(R1-1+R2-1)μ=μ1TR1-1μ1+μ2TR2-1μ2-(μ1-μ2)T(R1+R2)-1(μ1-μ2).
A=xT(R1-1+R2-1)x-2xT(R1-1+R2-1)μ+μT(R1-1+R2-1)μ+(μ1-μ2)T(R1+R2)-1(μ1-μ2)=(x-μ)T(R1-1+R2-1)(x-μ)+(μ1-μ2)T(R1+R2)-1(μ1-μ2).
β=exp-14(μ1-μ2)T(R1+R2)-1(μ1-μ2)(det R1)1/4(det R2)1/4×1(2π)N/2exp-14(x-μ)T(R1-1+R2-1)×(x-μ){[1+S1(x)][1+S2(x)]}1/2 dx=exp-14(μ1-μ2)T(R1+R2)-1(μ1-μ2)(det R)1/2(det R1)1/4(det R2)1/4
×1(2π)N/2(det R)1/2exp-12(x-μ)TR-1(x-μ)×{[1+S1(x)][1+S2(x)]}1/2 dx,
R-1=R1-1+R2-12.
(det R)1/2(det R1)1/4(det R2)14=(det R1)1/4(det R2)1/4(det R1)1/2(det R-1)1/2(det R2)1/2=(det R1R2)1/4detR1+R221/2.
β=exp-14(μ1-μ2)T(R1+R2)-1(μ1-μ2)(det R1R2)1/4detR1+R221/2E{[1+S1(x)][1+S2(x)]}1/2.
Q=exp-14(μ1-μ2)T(R1+R2)-1(μ1-μ2)×(det R1R2)1/4detR1+R221/2.
β=QE{[1+S1(x)][1+S2(x)]}1/2,
12QE{[1+S1(x)][1+S2(x)]}1/2,
Si(x)=m=3Ei{[Hm(Ri-1/2(X-μi)T)Hm(Ri-1/2(x-μi))]}=0,i=1,2.
β=Q,
12Q.
-logN(μ, R)1+n=3mE[HnT(R-1/2(X-μ))]×Hn(R-1/2(x-μ)).
f(x)=N(μ, R)1+n=3mE[HnT(Y)]Hn(y),
-logN(μ, R)1+n=3mE[HnT(Y)]Hn(y).
Dx*=mini-logN(μi, Ri)1+n=3mE[HnT(Ri-1/2(X-μi))]Hn(Ri-1/2(x-μi)).
Dx*>Tb,
i*=arg mini-logN(μi, Ri)×1+n=3mE[HnT(Ri-1/2(X-μi))]×Hn(R1-1/2(X-μi)).
Dx=-logN(μi*, Ri*)1+n=3mE[HnT(Ri*-1/2(X-μi))]×Hn(Ri*-1/2(x-μi*)).
1i*6,Dx<T,
μT(R1-1+R2-1)μ=μ1TR1-1μ1+μ2TR2-1μ2-(μ1-μ2)T(R1+R2)-1(μ1-μ2).
μT(R1-1+R2-1)μ=(R1-1μ1+R2-1μ2)T(R1-1+R2-1)-T(R1-1μ1+R2-1μ2).
μT(R1-1+R2-1)μ=μ1TR1-1(R1-1+R2-1)-1R1-1μ1+2μ1TR1-1(R1-1+R2-1)-1R2-1μ2+μ2TR2-1(R1-1+R2-1)-1R2-1μ2.
(R1-1+R2-1)-1=R1-R1(R1+R2)-1R1=R2-R2(R1+R2)-1R2.
μT(R1-1+R2-1)μ=μ1TR1-1μ1-μ1T(R1+R2)-1μ1+μ2TR2-1μ2-μ2T(R1+R2)-1μ2+2μ1TR1-1(R1-1+R2-1)-1×(-R1-1+R1-1+R2-1)μ2=μ1TR1-1μ1-μ1T(R1+R2)-1μ1+μ2TR2-1μ2-μ2T(R1+R2)-1μ2-2μ1TR1-1×[R1-R1(R1+R2)-1R1]R1-1μ2+2μ1TR1-1μ2=μ1TR1-1μ1-μ1T(R1+R2)-1μ1+μ2TR2-1μ2-μ2T(R1+R2)-1μ2+2μ1T(R1+R2)-1μ2, =μ1TR1-1μ1+μ2TR2-1μ2-(μ1-μ2)T×(R1+R2)-1(μ1-μ2),

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