Abstract

This paper describes a new kernel wavelet-based anomaly detection technique for long-wave (LW) forward-looking infrared imagery. The proposed approach called kernel wavelet-Reed–Xiaoli (wavelet-RX) algorithm is essentially an extension of the wavelet-RX algorithm (combination of wavelet transform and RX anomaly detector) to a high-dimensional feature space (possibly infinite) via a certain nonlinear mapping function of the input data. The wavelet-RX algorithm in this high-dimensional feature space can easily be implemented in terms of kernels that implicitly compute dot products in the feature space (kernelizing the wavelet-RX algorithm). In the proposed kernel wavelet-RX algorithm, a two-dimensional wavelet transform is first applied to decompose the input image into uniform subbands. A number of significant subbands (high-energy subbands) are concatenated together to form a subband-image cube. The kernel RX algorithm is then applied to this subband-image cube. Experimental results are presented for the proposed kernel wavelet-RX, wavelet-RX, and the classical constant false alarm rate (CFAR) algorithm for detecting anomalies (targets) in a large database of LW imagery. The receiver operating characteristic plots show that the proposed kernel wavelet-RX algorithm outperforms the wavelet-RX as well as the classical CFAR detector.

© 2011 Optical Society of America

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  1. D. W. J. Stein, S. G. Beaven, L. E. Hoff, E. M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Process. Mag. 19 (1), 58–69(2002).
    [CrossRef]
  2. H. Kwon, S. Z. Der, and N. M. Nasrabadi, “Adaptive anomaly detection using subspace separation for hyperspectral images,” Opt. Eng. 42, 3342–3351 (2003).
    [CrossRef]
  3. C. I. Chang and S. S. Chiang, “Anomaly detection and classification for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 40, 1314–1325 (2002).
    [CrossRef]
  4. S. Tiwari, S. Agarwal, and A. Trang, “FastKRX: a fast approximation for kernel RX anomaly detection,” Proc. SPIE 6953, 695316 (2008).
    [CrossRef]
  5. S. Kuttikkad and R. Chellappa, “Non-Gaussian CFAR techniques for target detection in high resolution SAR images,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 1994), pp. 910–914.
  6. B. Bhanu, “Automatic target recognition: state of the art survey,” IEEE Trans. Aerosp. Electron. Syst. AES-22, 364–379 (1986).
    [CrossRef]
  7. A. Margalit, I. S. Reed, and R. M. Gagliardi, “Adaptive optical target detection using correlated images,” IEEE Trans. Aerosp. Electron. Syst. AES-21, 394–405 (1985).
    [CrossRef]
  8. J. F. Khan and M. S. Alam, “Target detection in cluttered forward-looking infrared imagery,” Opt. Eng. 44, 076404 (2005).
    [CrossRef]
  9. J. F. Khan, M. S. Alam, and S. M. A. Bhuiyan, “Automatic target detection in forward-looking infrared imagery via probabilistic neural networks,” Appl. Opt. 48, 464–476 (2009).
    [CrossRef] [PubMed]
  10. S. Z. Der and R. Chellappa, “Probe-based automatic target recognition in infrared imagery,” IEEE Trans. Image Process. 6, 92–102 (1997).
    [CrossRef] [PubMed]
  11. B. V. K. V. Kumar, A. Mahalanobis, and R. D. Juday, Correlation Pattern Recognition (Cambridge University, 2005).
    [CrossRef]
  12. A. L. Chan, S. Z. Der, and N. M. Nasrabadi, “Multistage infrared target detection,” Opt. Eng. 42, 2746–2754 (2003).
    [CrossRef]
  13. M. Burton and C. Benning, “Comparison of imaging infrared detection algorithms,” Proc. SPIE 302, 26–32 (1981).
  14. A. Mehmood and N. M. Nasrabadi, “Wavelet-RX anomaly detection for dual-band forward-looking infrared imagery,” Appl. Opt. 49, 4621–4632 (2010).
    [CrossRef] [PubMed]
  15. I. S. Reed and X. Yu, “Adaptive multi-band CFAR detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust. Speech Signal Process. 38, 1760–1170(1990).
    [CrossRef]
  16. S. G. Mallat, A Wavelet Tour of Signal Processing, 2nd ed.(Academic, 2008).
  17. H. Kwon and N. M. Nasrabadi, “Kernel RX-algorithm: a nonlinear anomaly detector for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 43, 388–397 (2005).
    [CrossRef]
  18. E. Abdelkawy and D. McGaughy, “Wavelet-based image target detection methods,” Proc. SPIE 5094, 337–347 (2003).
    [CrossRef]
  19. S. Wen, Y. Jiangping, and L. Jian, “Small IR target detection approach based on multiscale relative distance image,” Proc. SPIE 4077, 194–197 (2000).
    [CrossRef]
  20. K. Riley and A. J. Devaney, “Wavelet processing of images for target detection,” Int. J. Imaging Syst. Technol. 7, 404–420(1996).
    [CrossRef]
  21. R. N. Strickland and H. L. Hahn, “Wavelet transforms methods for object detection and recovery,” IEEE Trans. Image Process. 6, 724–735 (1997).
    [CrossRef] [PubMed]
  22. X. Yu, I. S. Reed, W. Kraske, and A. D. Stocker, “A robust adaptive multi-spectral object detection by using wavelet transform,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1992), Vol.  99, pp. 141–144.
  23. B. Schölkopf and A. J. Samola, Learning with Kernels (MIT, 2002).

2010 (1)

2009 (1)

2008 (1)

S. Tiwari, S. Agarwal, and A. Trang, “FastKRX: a fast approximation for kernel RX anomaly detection,” Proc. SPIE 6953, 695316 (2008).
[CrossRef]

2005 (2)

J. F. Khan and M. S. Alam, “Target detection in cluttered forward-looking infrared imagery,” Opt. Eng. 44, 076404 (2005).
[CrossRef]

H. Kwon and N. M. Nasrabadi, “Kernel RX-algorithm: a nonlinear anomaly detector for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 43, 388–397 (2005).
[CrossRef]

2003 (3)

E. Abdelkawy and D. McGaughy, “Wavelet-based image target detection methods,” Proc. SPIE 5094, 337–347 (2003).
[CrossRef]

A. L. Chan, S. Z. Der, and N. M. Nasrabadi, “Multistage infrared target detection,” Opt. Eng. 42, 2746–2754 (2003).
[CrossRef]

H. Kwon, S. Z. Der, and N. M. Nasrabadi, “Adaptive anomaly detection using subspace separation for hyperspectral images,” Opt. Eng. 42, 3342–3351 (2003).
[CrossRef]

2002 (2)

C. I. Chang and S. S. Chiang, “Anomaly detection and classification for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 40, 1314–1325 (2002).
[CrossRef]

D. W. J. Stein, S. G. Beaven, L. E. Hoff, E. M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Process. Mag. 19 (1), 58–69(2002).
[CrossRef]

2000 (1)

S. Wen, Y. Jiangping, and L. Jian, “Small IR target detection approach based on multiscale relative distance image,” Proc. SPIE 4077, 194–197 (2000).
[CrossRef]

1997 (2)

S. Z. Der and R. Chellappa, “Probe-based automatic target recognition in infrared imagery,” IEEE Trans. Image Process. 6, 92–102 (1997).
[CrossRef] [PubMed]

R. N. Strickland and H. L. Hahn, “Wavelet transforms methods for object detection and recovery,” IEEE Trans. Image Process. 6, 724–735 (1997).
[CrossRef] [PubMed]

1996 (1)

K. Riley and A. J. Devaney, “Wavelet processing of images for target detection,” Int. J. Imaging Syst. Technol. 7, 404–420(1996).
[CrossRef]

1990 (1)

I. S. Reed and X. Yu, “Adaptive multi-band CFAR detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust. Speech Signal Process. 38, 1760–1170(1990).
[CrossRef]

1986 (1)

B. Bhanu, “Automatic target recognition: state of the art survey,” IEEE Trans. Aerosp. Electron. Syst. AES-22, 364–379 (1986).
[CrossRef]

1985 (1)

A. Margalit, I. S. Reed, and R. M. Gagliardi, “Adaptive optical target detection using correlated images,” IEEE Trans. Aerosp. Electron. Syst. AES-21, 394–405 (1985).
[CrossRef]

1981 (1)

M. Burton and C. Benning, “Comparison of imaging infrared detection algorithms,” Proc. SPIE 302, 26–32 (1981).

Abdelkawy, E.

E. Abdelkawy and D. McGaughy, “Wavelet-based image target detection methods,” Proc. SPIE 5094, 337–347 (2003).
[CrossRef]

Agarwal, S.

S. Tiwari, S. Agarwal, and A. Trang, “FastKRX: a fast approximation for kernel RX anomaly detection,” Proc. SPIE 6953, 695316 (2008).
[CrossRef]

Alam, M. S.

Beaven, S. G.

D. W. J. Stein, S. G. Beaven, L. E. Hoff, E. M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Process. Mag. 19 (1), 58–69(2002).
[CrossRef]

Benning, C.

M. Burton and C. Benning, “Comparison of imaging infrared detection algorithms,” Proc. SPIE 302, 26–32 (1981).

Bhanu, B.

B. Bhanu, “Automatic target recognition: state of the art survey,” IEEE Trans. Aerosp. Electron. Syst. AES-22, 364–379 (1986).
[CrossRef]

Bhuiyan, S. M. A.

Burton, M.

M. Burton and C. Benning, “Comparison of imaging infrared detection algorithms,” Proc. SPIE 302, 26–32 (1981).

Chan, A. L.

A. L. Chan, S. Z. Der, and N. M. Nasrabadi, “Multistage infrared target detection,” Opt. Eng. 42, 2746–2754 (2003).
[CrossRef]

Chang, C. I.

C. I. Chang and S. S. Chiang, “Anomaly detection and classification for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 40, 1314–1325 (2002).
[CrossRef]

Chellappa, R.

S. Z. Der and R. Chellappa, “Probe-based automatic target recognition in infrared imagery,” IEEE Trans. Image Process. 6, 92–102 (1997).
[CrossRef] [PubMed]

S. Kuttikkad and R. Chellappa, “Non-Gaussian CFAR techniques for target detection in high resolution SAR images,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 1994), pp. 910–914.

Chiang, S. S.

C. I. Chang and S. S. Chiang, “Anomaly detection and classification for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 40, 1314–1325 (2002).
[CrossRef]

Der, S. Z.

H. Kwon, S. Z. Der, and N. M. Nasrabadi, “Adaptive anomaly detection using subspace separation for hyperspectral images,” Opt. Eng. 42, 3342–3351 (2003).
[CrossRef]

A. L. Chan, S. Z. Der, and N. M. Nasrabadi, “Multistage infrared target detection,” Opt. Eng. 42, 2746–2754 (2003).
[CrossRef]

S. Z. Der and R. Chellappa, “Probe-based automatic target recognition in infrared imagery,” IEEE Trans. Image Process. 6, 92–102 (1997).
[CrossRef] [PubMed]

Devaney, A. J.

K. Riley and A. J. Devaney, “Wavelet processing of images for target detection,” Int. J. Imaging Syst. Technol. 7, 404–420(1996).
[CrossRef]

Gagliardi, R. M.

A. Margalit, I. S. Reed, and R. M. Gagliardi, “Adaptive optical target detection using correlated images,” IEEE Trans. Aerosp. Electron. Syst. AES-21, 394–405 (1985).
[CrossRef]

Hahn, H. L.

R. N. Strickland and H. L. Hahn, “Wavelet transforms methods for object detection and recovery,” IEEE Trans. Image Process. 6, 724–735 (1997).
[CrossRef] [PubMed]

Hoff, L. E.

D. W. J. Stein, S. G. Beaven, L. E. Hoff, E. M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Process. Mag. 19 (1), 58–69(2002).
[CrossRef]

Jian, L.

S. Wen, Y. Jiangping, and L. Jian, “Small IR target detection approach based on multiscale relative distance image,” Proc. SPIE 4077, 194–197 (2000).
[CrossRef]

Jiangping, Y.

S. Wen, Y. Jiangping, and L. Jian, “Small IR target detection approach based on multiscale relative distance image,” Proc. SPIE 4077, 194–197 (2000).
[CrossRef]

Juday, R. D.

B. V. K. V. Kumar, A. Mahalanobis, and R. D. Juday, Correlation Pattern Recognition (Cambridge University, 2005).
[CrossRef]

Khan, J. F.

Kraske, W.

X. Yu, I. S. Reed, W. Kraske, and A. D. Stocker, “A robust adaptive multi-spectral object detection by using wavelet transform,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1992), Vol.  99, pp. 141–144.

Kumar, B. V. K. V.

B. V. K. V. Kumar, A. Mahalanobis, and R. D. Juday, Correlation Pattern Recognition (Cambridge University, 2005).
[CrossRef]

Kuttikkad, S.

S. Kuttikkad and R. Chellappa, “Non-Gaussian CFAR techniques for target detection in high resolution SAR images,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 1994), pp. 910–914.

Kwon, H.

H. Kwon and N. M. Nasrabadi, “Kernel RX-algorithm: a nonlinear anomaly detector for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 43, 388–397 (2005).
[CrossRef]

H. Kwon, S. Z. Der, and N. M. Nasrabadi, “Adaptive anomaly detection using subspace separation for hyperspectral images,” Opt. Eng. 42, 3342–3351 (2003).
[CrossRef]

Mahalanobis, A.

B. V. K. V. Kumar, A. Mahalanobis, and R. D. Juday, Correlation Pattern Recognition (Cambridge University, 2005).
[CrossRef]

Mallat, S. G.

S. G. Mallat, A Wavelet Tour of Signal Processing, 2nd ed.(Academic, 2008).

Margalit, A.

A. Margalit, I. S. Reed, and R. M. Gagliardi, “Adaptive optical target detection using correlated images,” IEEE Trans. Aerosp. Electron. Syst. AES-21, 394–405 (1985).
[CrossRef]

McGaughy, D.

E. Abdelkawy and D. McGaughy, “Wavelet-based image target detection methods,” Proc. SPIE 5094, 337–347 (2003).
[CrossRef]

Mehmood, A.

Nasrabadi, N. M.

A. Mehmood and N. M. Nasrabadi, “Wavelet-RX anomaly detection for dual-band forward-looking infrared imagery,” Appl. Opt. 49, 4621–4632 (2010).
[CrossRef] [PubMed]

H. Kwon and N. M. Nasrabadi, “Kernel RX-algorithm: a nonlinear anomaly detector for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 43, 388–397 (2005).
[CrossRef]

H. Kwon, S. Z. Der, and N. M. Nasrabadi, “Adaptive anomaly detection using subspace separation for hyperspectral images,” Opt. Eng. 42, 3342–3351 (2003).
[CrossRef]

A. L. Chan, S. Z. Der, and N. M. Nasrabadi, “Multistage infrared target detection,” Opt. Eng. 42, 2746–2754 (2003).
[CrossRef]

Reed, I. S.

I. S. Reed and X. Yu, “Adaptive multi-band CFAR detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust. Speech Signal Process. 38, 1760–1170(1990).
[CrossRef]

A. Margalit, I. S. Reed, and R. M. Gagliardi, “Adaptive optical target detection using correlated images,” IEEE Trans. Aerosp. Electron. Syst. AES-21, 394–405 (1985).
[CrossRef]

X. Yu, I. S. Reed, W. Kraske, and A. D. Stocker, “A robust adaptive multi-spectral object detection by using wavelet transform,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1992), Vol.  99, pp. 141–144.

Riley, K.

K. Riley and A. J. Devaney, “Wavelet processing of images for target detection,” Int. J. Imaging Syst. Technol. 7, 404–420(1996).
[CrossRef]

Samola, A. J.

B. Schölkopf and A. J. Samola, Learning with Kernels (MIT, 2002).

Schaum, A. P.

D. W. J. Stein, S. G. Beaven, L. E. Hoff, E. M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Process. Mag. 19 (1), 58–69(2002).
[CrossRef]

Schölkopf, B.

B. Schölkopf and A. J. Samola, Learning with Kernels (MIT, 2002).

Stein, D. W. J.

D. W. J. Stein, S. G. Beaven, L. E. Hoff, E. M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Process. Mag. 19 (1), 58–69(2002).
[CrossRef]

Stocker, A. D.

D. W. J. Stein, S. G. Beaven, L. E. Hoff, E. M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Process. Mag. 19 (1), 58–69(2002).
[CrossRef]

X. Yu, I. S. Reed, W. Kraske, and A. D. Stocker, “A robust adaptive multi-spectral object detection by using wavelet transform,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1992), Vol.  99, pp. 141–144.

Strickland, R. N.

R. N. Strickland and H. L. Hahn, “Wavelet transforms methods for object detection and recovery,” IEEE Trans. Image Process. 6, 724–735 (1997).
[CrossRef] [PubMed]

Tiwari, S.

S. Tiwari, S. Agarwal, and A. Trang, “FastKRX: a fast approximation for kernel RX anomaly detection,” Proc. SPIE 6953, 695316 (2008).
[CrossRef]

Trang, A.

S. Tiwari, S. Agarwal, and A. Trang, “FastKRX: a fast approximation for kernel RX anomaly detection,” Proc. SPIE 6953, 695316 (2008).
[CrossRef]

Wen, S.

S. Wen, Y. Jiangping, and L. Jian, “Small IR target detection approach based on multiscale relative distance image,” Proc. SPIE 4077, 194–197 (2000).
[CrossRef]

Winter, E. M.

D. W. J. Stein, S. G. Beaven, L. E. Hoff, E. M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Process. Mag. 19 (1), 58–69(2002).
[CrossRef]

Yu, X.

I. S. Reed and X. Yu, “Adaptive multi-band CFAR detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust. Speech Signal Process. 38, 1760–1170(1990).
[CrossRef]

X. Yu, I. S. Reed, W. Kraske, and A. D. Stocker, “A robust adaptive multi-spectral object detection by using wavelet transform,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1992), Vol.  99, pp. 141–144.

Appl. Opt. (2)

IEEE Signal Process. Mag. (1)

D. W. J. Stein, S. G. Beaven, L. E. Hoff, E. M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Process. Mag. 19 (1), 58–69(2002).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

I. S. Reed and X. Yu, “Adaptive multi-band CFAR detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust. Speech Signal Process. 38, 1760–1170(1990).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst. (2)

B. Bhanu, “Automatic target recognition: state of the art survey,” IEEE Trans. Aerosp. Electron. Syst. AES-22, 364–379 (1986).
[CrossRef]

A. Margalit, I. S. Reed, and R. M. Gagliardi, “Adaptive optical target detection using correlated images,” IEEE Trans. Aerosp. Electron. Syst. AES-21, 394–405 (1985).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (2)

C. I. Chang and S. S. Chiang, “Anomaly detection and classification for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 40, 1314–1325 (2002).
[CrossRef]

H. Kwon and N. M. Nasrabadi, “Kernel RX-algorithm: a nonlinear anomaly detector for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 43, 388–397 (2005).
[CrossRef]

IEEE Trans. Image Process. (2)

R. N. Strickland and H. L. Hahn, “Wavelet transforms methods for object detection and recovery,” IEEE Trans. Image Process. 6, 724–735 (1997).
[CrossRef] [PubMed]

S. Z. Der and R. Chellappa, “Probe-based automatic target recognition in infrared imagery,” IEEE Trans. Image Process. 6, 92–102 (1997).
[CrossRef] [PubMed]

Int. J. Imaging Syst. Technol. (1)

K. Riley and A. J. Devaney, “Wavelet processing of images for target detection,” Int. J. Imaging Syst. Technol. 7, 404–420(1996).
[CrossRef]

Opt. Eng. (3)

J. F. Khan and M. S. Alam, “Target detection in cluttered forward-looking infrared imagery,” Opt. Eng. 44, 076404 (2005).
[CrossRef]

H. Kwon, S. Z. Der, and N. M. Nasrabadi, “Adaptive anomaly detection using subspace separation for hyperspectral images,” Opt. Eng. 42, 3342–3351 (2003).
[CrossRef]

A. L. Chan, S. Z. Der, and N. M. Nasrabadi, “Multistage infrared target detection,” Opt. Eng. 42, 2746–2754 (2003).
[CrossRef]

Proc. SPIE (4)

M. Burton and C. Benning, “Comparison of imaging infrared detection algorithms,” Proc. SPIE 302, 26–32 (1981).

E. Abdelkawy and D. McGaughy, “Wavelet-based image target detection methods,” Proc. SPIE 5094, 337–347 (2003).
[CrossRef]

S. Wen, Y. Jiangping, and L. Jian, “Small IR target detection approach based on multiscale relative distance image,” Proc. SPIE 4077, 194–197 (2000).
[CrossRef]

S. Tiwari, S. Agarwal, and A. Trang, “FastKRX: a fast approximation for kernel RX anomaly detection,” Proc. SPIE 6953, 695316 (2008).
[CrossRef]

Other (5)

S. Kuttikkad and R. Chellappa, “Non-Gaussian CFAR techniques for target detection in high resolution SAR images,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 1994), pp. 910–914.

B. V. K. V. Kumar, A. Mahalanobis, and R. D. Juday, Correlation Pattern Recognition (Cambridge University, 2005).
[CrossRef]

X. Yu, I. S. Reed, W. Kraske, and A. D. Stocker, “A robust adaptive multi-spectral object detection by using wavelet transform,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1992), Vol.  99, pp. 141–144.

B. Schölkopf and A. J. Samola, Learning with Kernels (MIT, 2002).

S. G. Mallat, A Wavelet Tour of Signal Processing, 2nd ed.(Academic, 2008).

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Figures (5)

Fig. 1
Fig. 1

Sliding dual window: an IWR and an OWR.

Fig. 2
Fig. 2

Wavelet subband-image cube obtained by a uniform wavelet decomposition of the original signal.

Fig. 3
Fig. 3

Uniform three-level wavelet decomposition of an LW image.

Fig. 4
Fig. 4

(a) LW raw image, (b) CFAR, (c) WRX, and (d) KWRX.

Fig. 5
Fig. 5

ROC plots of (a) KWRX applied to different bands; (b) KWRX applied to five LW bands with different σ values; (c) CFAR, WRX, and KWRX applied to LW.

Equations (20)

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δ CFAR ( r i j ) = ( r i j μ ) T σ 1 ( r i j μ ) ,
x ( i , j ) = ( x 1 ( i , j ) , x 2 ( i , j ) , , x B ( i , j ) ) T ,
X = [ x ( 1 ) , x ( 2 ) , , x ( N ) ] .
δ WRX ( r i j ) = ( r i j μ ^ ) T C ^ 1 ( r i j μ ^ ) ,
Φ χ F , x Φ ( x ) ,
k ( x i , x j ) = Φ ( x i ) , Φ ( x j ) = Φ ( x i ) · Φ ( x j ) .
δ WRX ( Φ ( r ) ) = ( Φ ( r ) μ ^ Φ ) T C ^ Φ 1 ( Φ ( r ) μ ^ Φ ) ,
C ^ Φ = 1 N i = 1 N ( Φ ( x ( i ) ) μ ^ Φ ) ( Φ ( x ( i ) ) μ ^ Φ ) T ,
μ ^ Φ = 1 N i = 1 N ( Φ ( x ( i ) ) ) .
C ^ ϕ = V Φ Λ V Φ T ,
V Φ = [ v Φ 1 , v Φ 2 , v Φ 3 , ] ,
C ^ ϕ # = V Φ Λ 1 V Φ T .
v Φ j = i = 1 N β i j Φ c ( x ( i ) ) = X Φ β j ,
V Φ = X Φ B ,
C ^ ϕ 1 = X Φ B Λ 1 B T X Φ T .
δ KWRX ( Φ ( r ) ) = ( Φ ( r ) μ ^ Φ ) T X Φ B Λ 1 B T X Φ T ( Φ ( r ) μ ^ Φ ) .
Φ ( r ) T X Φ = Φ ( r ) T ( [ Φ ( x ( 1 ) ) Φ ( x ( 2 ) ) Φ ( x ( 3 ) ) Φ ( x ( N ) ) ] 1 N i = 1 N Φ ( x ( i ) ) ) = ( k ( x ( 1 ) , r ) k ( x ( 2 ) , r ) k ( x ( 3 ) , r ) k ( x ( N ) , r ) ) 1 N i = 1 N k ( x ( i ) , r ) = k ( X , r ) T 1 N i = 1 N k ( x ( i ) , r ) k r T ,
μ ^ Φ T X Φ = 1 N i = 1 N Φ ( x ( i ) ) T × [ ( Φ ( x ( 1 ) ) Φ ( x ( 2 ) ) Φ ( x ( 3 ) ) Φ ( x ( N ) ) ) 1 N i = 1 N Φ ( x ( i ) ) ] = 1 N i = 1 N ( k ( x ( i ) , x ( 1 ) ) × k ( x ( i ) , x ( 2 ) ) k ( x ( i ) , x ( 3 ) ) k ( x ( i ) , x ( N ) ) ) 1 N 2 i = 1 N j = 1 N k ( x ( i ) , x ( j ) ) = 1 N i = 1 N k ( x ( i ) , x ) 1 N 2 i = 1 N j = 1 N k ( x ( i ) , x ( j ) ) k μ ^ T .
K ^ 1 = 1 N B Λ 1 B T .
δ KWRX ( k r ) = ( k r k μ ^ ) T K ^ 1 ( k r k μ ^ ) ,

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