Abstract

Millimeter wave imaging is finding rapid adoption in security applications such as the detection of objects concealed under clothing. A passive imaging system can be realized as a stand-off type sensor that can operate in open spaces, both indoors and outdoors. In this paper, we address real-time outdoor concealed-object detection and segmentation with a radiometric imaging system operating in the W-band. The imaging system is equipped with a dielectric lens and a receiver array operating at around 94 GHz. Images are analyzed by multilevel segmentation to identify a concealed object. Each level of segmentation comprises vector quantization, expectation-maximization, and Bayesian decision making to cluster pixels on the basis of a Gaussian mixture model. In addition, we describe a faster process that adopts only vector quantization for the first level segmentation. Experiments confirm that the proposed methods provide fast and reliable detection and segmentation for a moving human subject carrying a concealed gun.

© 2011 OSA

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    [CrossRef] [PubMed]
  6. F. Buersgens, G. Acuna, and R. Kersting, “Millimeter wave probing of the acoustic phase for concealed object detection,” Opt. Express 15(14), 8838–8843 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-14-8838 .
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2010 (3)

H. Sato, K. Sawaya, K. Mizuno, J. Uemura, M. Takeda, J. Takahashi, K. Yamada, K. Morichika, T. Hasegawa, H. Hirai, H. Niikura, T. Matsuzaki, S. Kato, and J. Nakada, “Passive millimeter-wave imaging for security and safety applications,” Proc. SPIE 7672, 76720V (2010).

D.-S. Lee, S. Yeom, J. Y. Son, and S.-H. Kim, “Automatic image segmentation for concealed object detection using the expectation-maximization algorithm,” Opt. Express 18(10), 10659–10667 (2010).
[CrossRef] [PubMed]

S. Yeom, D.-S. Lee, H. Lee, J. Son, and V. P. Guschin, “Three-dimensional passive millimeter-wave imaging and depth estimation,” Proc. SPIE 7690, 76900W, 76900W-6 (2010).
[CrossRef]

2009 (1)

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, I. Mehdi, and P. H. Siegel, “An approach for sub-second imaging of concealed objects using terahertz (THz) radar,”J. Infrared, Millimeter Terahertz Waves 30, 1297–1307 (2009).

2008 (2)

X. Shen, C. R. Dietlein, E. Grossman, Z. Popovic, and F. G. Meyer, “Detection and segmentation of concealed objects in terahertz images,” IEEE Trans. Image Process. 17(12), 2465–2475 (2008).
[CrossRef] [PubMed]

M. R. Fetterman, J. Grata, G. Jubic, W. L. Kiser, and A. Visnansky, “Simulation, acquisition and analysis of passive millimeter-wave images in remote sensing applications,” Opt. Express 16(25), 20503–20515 (2008).
[CrossRef] [PubMed]

2007 (3)

F. Buersgens, G. Acuna, and R. Kersting, “Millimeter wave probing of the acoustic phase for concealed object detection,” Opt. Express 15(14), 8838–8843 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-14-8838 .
[CrossRef] [PubMed]

C. D. Haworth, Y. de Saint-Pern, D. Clark, E. Trucco, and Y. R. Petillot, “Detection and tracking of multiple metallic objects in millimeter-wave images,” Int. J. Comput. Vis. 71(2), 183–196 (2007).
[CrossRef]

R. Appleby and R. N. Anderton, “Millimeter-wave and submillimeter-wave imaging for security and surveillance,” Proc. IEEE 95(8), 1683–1690 (2007).
[CrossRef]

2005 (1)

H.-M. Chen, S. Lee, R. M. Rao, M.-A. Slamani, and P. K. Varshney, “Imaging for concealed weapon detection: a tutorial overview of development in imaging sensors and processing,” IEEE Signal Process. Mag. 22(2), 52–61 (2005).
[CrossRef]

2003 (1)

L. Yujiri, M. Shoucri, and P. Moffa, “Passive millimeter-wave imaging,” IEEE Microw. Mag. 4(3), 39–50 (2003).
[CrossRef]

Acuna, G.

F. Buersgens, G. Acuna, and R. Kersting, “Millimeter wave probing of the acoustic phase for concealed object detection,” Opt. Express 15(14), 8838–8843 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-14-8838 .
[CrossRef] [PubMed]

Anderton, R. N.

R. Appleby and R. N. Anderton, “Millimeter-wave and submillimeter-wave imaging for security and surveillance,” Proc. IEEE 95(8), 1683–1690 (2007).
[CrossRef]

Appleby, R.

R. Appleby and R. N. Anderton, “Millimeter-wave and submillimeter-wave imaging for security and surveillance,” Proc. IEEE 95(8), 1683–1690 (2007).
[CrossRef]

Bryllert, T.

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, I. Mehdi, and P. H. Siegel, “An approach for sub-second imaging of concealed objects using terahertz (THz) radar,”J. Infrared, Millimeter Terahertz Waves 30, 1297–1307 (2009).

Buersgens, F.

F. Buersgens, G. Acuna, and R. Kersting, “Millimeter wave probing of the acoustic phase for concealed object detection,” Opt. Express 15(14), 8838–8843 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-14-8838 .
[CrossRef] [PubMed]

Chattopadhyay, G.

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, I. Mehdi, and P. H. Siegel, “An approach for sub-second imaging of concealed objects using terahertz (THz) radar,”J. Infrared, Millimeter Terahertz Waves 30, 1297–1307 (2009).

Chen, H.-M.

H.-M. Chen, S. Lee, R. M. Rao, M.-A. Slamani, and P. K. Varshney, “Imaging for concealed weapon detection: a tutorial overview of development in imaging sensors and processing,” IEEE Signal Process. Mag. 22(2), 52–61 (2005).
[CrossRef]

Clark, D.

C. D. Haworth, Y. de Saint-Pern, D. Clark, E. Trucco, and Y. R. Petillot, “Detection and tracking of multiple metallic objects in millimeter-wave images,” Int. J. Comput. Vis. 71(2), 183–196 (2007).
[CrossRef]

Cooper, K. B.

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, I. Mehdi, and P. H. Siegel, “An approach for sub-second imaging of concealed objects using terahertz (THz) radar,”J. Infrared, Millimeter Terahertz Waves 30, 1297–1307 (2009).

de Saint-Pern, Y.

C. D. Haworth, Y. de Saint-Pern, D. Clark, E. Trucco, and Y. R. Petillot, “Detection and tracking of multiple metallic objects in millimeter-wave images,” Int. J. Comput. Vis. 71(2), 183–196 (2007).
[CrossRef]

Dengler, R. J.

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, I. Mehdi, and P. H. Siegel, “An approach for sub-second imaging of concealed objects using terahertz (THz) radar,”J. Infrared, Millimeter Terahertz Waves 30, 1297–1307 (2009).

Dietlein, C. R.

X. Shen, C. R. Dietlein, E. Grossman, Z. Popovic, and F. G. Meyer, “Detection and segmentation of concealed objects in terahertz images,” IEEE Trans. Image Process. 17(12), 2465–2475 (2008).
[CrossRef] [PubMed]

Fetterman, M. R.

M. R. Fetterman, J. Grata, G. Jubic, W. L. Kiser, and A. Visnansky, “Simulation, acquisition and analysis of passive millimeter-wave images in remote sensing applications,” Opt. Express 16(25), 20503–20515 (2008).
[CrossRef] [PubMed]

Grata, J.

M. R. Fetterman, J. Grata, G. Jubic, W. L. Kiser, and A. Visnansky, “Simulation, acquisition and analysis of passive millimeter-wave images in remote sensing applications,” Opt. Express 16(25), 20503–20515 (2008).
[CrossRef] [PubMed]

Grossman, E.

X. Shen, C. R. Dietlein, E. Grossman, Z. Popovic, and F. G. Meyer, “Detection and segmentation of concealed objects in terahertz images,” IEEE Trans. Image Process. 17(12), 2465–2475 (2008).
[CrossRef] [PubMed]

Guschin, V. P.

S. Yeom, D.-S. Lee, H. Lee, J. Son, and V. P. Guschin, “Three-dimensional passive millimeter-wave imaging and depth estimation,” Proc. SPIE 7690, 76900W, 76900W-6 (2010).
[CrossRef]

Hasegawa, T.

H. Sato, K. Sawaya, K. Mizuno, J. Uemura, M. Takeda, J. Takahashi, K. Yamada, K. Morichika, T. Hasegawa, H. Hirai, H. Niikura, T. Matsuzaki, S. Kato, and J. Nakada, “Passive millimeter-wave imaging for security and safety applications,” Proc. SPIE 7672, 76720V (2010).

Haworth, C. D.

C. D. Haworth, Y. de Saint-Pern, D. Clark, E. Trucco, and Y. R. Petillot, “Detection and tracking of multiple metallic objects in millimeter-wave images,” Int. J. Comput. Vis. 71(2), 183–196 (2007).
[CrossRef]

Hirai, H.

H. Sato, K. Sawaya, K. Mizuno, J. Uemura, M. Takeda, J. Takahashi, K. Yamada, K. Morichika, T. Hasegawa, H. Hirai, H. Niikura, T. Matsuzaki, S. Kato, and J. Nakada, “Passive millimeter-wave imaging for security and safety applications,” Proc. SPIE 7672, 76720V (2010).

Jubic, G.

M. R. Fetterman, J. Grata, G. Jubic, W. L. Kiser, and A. Visnansky, “Simulation, acquisition and analysis of passive millimeter-wave images in remote sensing applications,” Opt. Express 16(25), 20503–20515 (2008).
[CrossRef] [PubMed]

Kato, S.

H. Sato, K. Sawaya, K. Mizuno, J. Uemura, M. Takeda, J. Takahashi, K. Yamada, K. Morichika, T. Hasegawa, H. Hirai, H. Niikura, T. Matsuzaki, S. Kato, and J. Nakada, “Passive millimeter-wave imaging for security and safety applications,” Proc. SPIE 7672, 76720V (2010).

Kersting, R.

F. Buersgens, G. Acuna, and R. Kersting, “Millimeter wave probing of the acoustic phase for concealed object detection,” Opt. Express 15(14), 8838–8843 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-14-8838 .
[CrossRef] [PubMed]

Kim, S.-H.

D.-S. Lee, S. Yeom, J. Y. Son, and S.-H. Kim, “Automatic image segmentation for concealed object detection using the expectation-maximization algorithm,” Opt. Express 18(10), 10659–10667 (2010).
[CrossRef] [PubMed]

Kiser, W. L.

M. R. Fetterman, J. Grata, G. Jubic, W. L. Kiser, and A. Visnansky, “Simulation, acquisition and analysis of passive millimeter-wave images in remote sensing applications,” Opt. Express 16(25), 20503–20515 (2008).
[CrossRef] [PubMed]

Lee, D.-S.

S. Yeom, D.-S. Lee, H. Lee, J. Son, and V. P. Guschin, “Three-dimensional passive millimeter-wave imaging and depth estimation,” Proc. SPIE 7690, 76900W, 76900W-6 (2010).
[CrossRef]

D.-S. Lee, S. Yeom, J. Y. Son, and S.-H. Kim, “Automatic image segmentation for concealed object detection using the expectation-maximization algorithm,” Opt. Express 18(10), 10659–10667 (2010).
[CrossRef] [PubMed]

Lee, H.

S. Yeom, D.-S. Lee, H. Lee, J. Son, and V. P. Guschin, “Three-dimensional passive millimeter-wave imaging and depth estimation,” Proc. SPIE 7690, 76900W, 76900W-6 (2010).
[CrossRef]

Lee, S.

H.-M. Chen, S. Lee, R. M. Rao, M.-A. Slamani, and P. K. Varshney, “Imaging for concealed weapon detection: a tutorial overview of development in imaging sensors and processing,” IEEE Signal Process. Mag. 22(2), 52–61 (2005).
[CrossRef]

Llombart, N.

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, I. Mehdi, and P. H. Siegel, “An approach for sub-second imaging of concealed objects using terahertz (THz) radar,”J. Infrared, Millimeter Terahertz Waves 30, 1297–1307 (2009).

Matsuzaki, T.

H. Sato, K. Sawaya, K. Mizuno, J. Uemura, M. Takeda, J. Takahashi, K. Yamada, K. Morichika, T. Hasegawa, H. Hirai, H. Niikura, T. Matsuzaki, S. Kato, and J. Nakada, “Passive millimeter-wave imaging for security and safety applications,” Proc. SPIE 7672, 76720V (2010).

Mehdi, I.

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, I. Mehdi, and P. H. Siegel, “An approach for sub-second imaging of concealed objects using terahertz (THz) radar,”J. Infrared, Millimeter Terahertz Waves 30, 1297–1307 (2009).

Meyer, F. G.

X. Shen, C. R. Dietlein, E. Grossman, Z. Popovic, and F. G. Meyer, “Detection and segmentation of concealed objects in terahertz images,” IEEE Trans. Image Process. 17(12), 2465–2475 (2008).
[CrossRef] [PubMed]

Mizuno, K.

H. Sato, K. Sawaya, K. Mizuno, J. Uemura, M. Takeda, J. Takahashi, K. Yamada, K. Morichika, T. Hasegawa, H. Hirai, H. Niikura, T. Matsuzaki, S. Kato, and J. Nakada, “Passive millimeter-wave imaging for security and safety applications,” Proc. SPIE 7672, 76720V (2010).

Moffa, P.

L. Yujiri, M. Shoucri, and P. Moffa, “Passive millimeter-wave imaging,” IEEE Microw. Mag. 4(3), 39–50 (2003).
[CrossRef]

Morichika, K.

H. Sato, K. Sawaya, K. Mizuno, J. Uemura, M. Takeda, J. Takahashi, K. Yamada, K. Morichika, T. Hasegawa, H. Hirai, H. Niikura, T. Matsuzaki, S. Kato, and J. Nakada, “Passive millimeter-wave imaging for security and safety applications,” Proc. SPIE 7672, 76720V (2010).

Nakada, J.

H. Sato, K. Sawaya, K. Mizuno, J. Uemura, M. Takeda, J. Takahashi, K. Yamada, K. Morichika, T. Hasegawa, H. Hirai, H. Niikura, T. Matsuzaki, S. Kato, and J. Nakada, “Passive millimeter-wave imaging for security and safety applications,” Proc. SPIE 7672, 76720V (2010).

Niikura, H.

H. Sato, K. Sawaya, K. Mizuno, J. Uemura, M. Takeda, J. Takahashi, K. Yamada, K. Morichika, T. Hasegawa, H. Hirai, H. Niikura, T. Matsuzaki, S. Kato, and J. Nakada, “Passive millimeter-wave imaging for security and safety applications,” Proc. SPIE 7672, 76720V (2010).

Petillot, Y. R.

C. D. Haworth, Y. de Saint-Pern, D. Clark, E. Trucco, and Y. R. Petillot, “Detection and tracking of multiple metallic objects in millimeter-wave images,” Int. J. Comput. Vis. 71(2), 183–196 (2007).
[CrossRef]

Popovic, Z.

X. Shen, C. R. Dietlein, E. Grossman, Z. Popovic, and F. G. Meyer, “Detection and segmentation of concealed objects in terahertz images,” IEEE Trans. Image Process. 17(12), 2465–2475 (2008).
[CrossRef] [PubMed]

Rao, R. M.

H.-M. Chen, S. Lee, R. M. Rao, M.-A. Slamani, and P. K. Varshney, “Imaging for concealed weapon detection: a tutorial overview of development in imaging sensors and processing,” IEEE Signal Process. Mag. 22(2), 52–61 (2005).
[CrossRef]

Sato, H.

H. Sato, K. Sawaya, K. Mizuno, J. Uemura, M. Takeda, J. Takahashi, K. Yamada, K. Morichika, T. Hasegawa, H. Hirai, H. Niikura, T. Matsuzaki, S. Kato, and J. Nakada, “Passive millimeter-wave imaging for security and safety applications,” Proc. SPIE 7672, 76720V (2010).

Sawaya, K.

H. Sato, K. Sawaya, K. Mizuno, J. Uemura, M. Takeda, J. Takahashi, K. Yamada, K. Morichika, T. Hasegawa, H. Hirai, H. Niikura, T. Matsuzaki, S. Kato, and J. Nakada, “Passive millimeter-wave imaging for security and safety applications,” Proc. SPIE 7672, 76720V (2010).

Shen, X.

X. Shen, C. R. Dietlein, E. Grossman, Z. Popovic, and F. G. Meyer, “Detection and segmentation of concealed objects in terahertz images,” IEEE Trans. Image Process. 17(12), 2465–2475 (2008).
[CrossRef] [PubMed]

Shoucri, M.

L. Yujiri, M. Shoucri, and P. Moffa, “Passive millimeter-wave imaging,” IEEE Microw. Mag. 4(3), 39–50 (2003).
[CrossRef]

Siegel, P. H.

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, I. Mehdi, and P. H. Siegel, “An approach for sub-second imaging of concealed objects using terahertz (THz) radar,”J. Infrared, Millimeter Terahertz Waves 30, 1297–1307 (2009).

Slamani, M.-A.

H.-M. Chen, S. Lee, R. M. Rao, M.-A. Slamani, and P. K. Varshney, “Imaging for concealed weapon detection: a tutorial overview of development in imaging sensors and processing,” IEEE Signal Process. Mag. 22(2), 52–61 (2005).
[CrossRef]

Son, J.

S. Yeom, D.-S. Lee, H. Lee, J. Son, and V. P. Guschin, “Three-dimensional passive millimeter-wave imaging and depth estimation,” Proc. SPIE 7690, 76900W, 76900W-6 (2010).
[CrossRef]

Son, J. Y.

D.-S. Lee, S. Yeom, J. Y. Son, and S.-H. Kim, “Automatic image segmentation for concealed object detection using the expectation-maximization algorithm,” Opt. Express 18(10), 10659–10667 (2010).
[CrossRef] [PubMed]

Takahashi, J.

H. Sato, K. Sawaya, K. Mizuno, J. Uemura, M. Takeda, J. Takahashi, K. Yamada, K. Morichika, T. Hasegawa, H. Hirai, H. Niikura, T. Matsuzaki, S. Kato, and J. Nakada, “Passive millimeter-wave imaging for security and safety applications,” Proc. SPIE 7672, 76720V (2010).

Takeda, M.

H. Sato, K. Sawaya, K. Mizuno, J. Uemura, M. Takeda, J. Takahashi, K. Yamada, K. Morichika, T. Hasegawa, H. Hirai, H. Niikura, T. Matsuzaki, S. Kato, and J. Nakada, “Passive millimeter-wave imaging for security and safety applications,” Proc. SPIE 7672, 76720V (2010).

Trucco, E.

C. D. Haworth, Y. de Saint-Pern, D. Clark, E. Trucco, and Y. R. Petillot, “Detection and tracking of multiple metallic objects in millimeter-wave images,” Int. J. Comput. Vis. 71(2), 183–196 (2007).
[CrossRef]

Uemura, J.

H. Sato, K. Sawaya, K. Mizuno, J. Uemura, M. Takeda, J. Takahashi, K. Yamada, K. Morichika, T. Hasegawa, H. Hirai, H. Niikura, T. Matsuzaki, S. Kato, and J. Nakada, “Passive millimeter-wave imaging for security and safety applications,” Proc. SPIE 7672, 76720V (2010).

Varshney, P. K.

H.-M. Chen, S. Lee, R. M. Rao, M.-A. Slamani, and P. K. Varshney, “Imaging for concealed weapon detection: a tutorial overview of development in imaging sensors and processing,” IEEE Signal Process. Mag. 22(2), 52–61 (2005).
[CrossRef]

Visnansky, A.

M. R. Fetterman, J. Grata, G. Jubic, W. L. Kiser, and A. Visnansky, “Simulation, acquisition and analysis of passive millimeter-wave images in remote sensing applications,” Opt. Express 16(25), 20503–20515 (2008).
[CrossRef] [PubMed]

Yamada, K.

H. Sato, K. Sawaya, K. Mizuno, J. Uemura, M. Takeda, J. Takahashi, K. Yamada, K. Morichika, T. Hasegawa, H. Hirai, H. Niikura, T. Matsuzaki, S. Kato, and J. Nakada, “Passive millimeter-wave imaging for security and safety applications,” Proc. SPIE 7672, 76720V (2010).

Yeom, S.

D.-S. Lee, S. Yeom, J. Y. Son, and S.-H. Kim, “Automatic image segmentation for concealed object detection using the expectation-maximization algorithm,” Opt. Express 18(10), 10659–10667 (2010).
[CrossRef] [PubMed]

S. Yeom, D.-S. Lee, H. Lee, J. Son, and V. P. Guschin, “Three-dimensional passive millimeter-wave imaging and depth estimation,” Proc. SPIE 7690, 76900W, 76900W-6 (2010).
[CrossRef]

Yujiri, L.

L. Yujiri, M. Shoucri, and P. Moffa, “Passive millimeter-wave imaging,” IEEE Microw. Mag. 4(3), 39–50 (2003).
[CrossRef]

IEEE Microw. Mag. (1)

L. Yujiri, M. Shoucri, and P. Moffa, “Passive millimeter-wave imaging,” IEEE Microw. Mag. 4(3), 39–50 (2003).
[CrossRef]

IEEE Signal Process. Mag. (1)

H.-M. Chen, S. Lee, R. M. Rao, M.-A. Slamani, and P. K. Varshney, “Imaging for concealed weapon detection: a tutorial overview of development in imaging sensors and processing,” IEEE Signal Process. Mag. 22(2), 52–61 (2005).
[CrossRef]

IEEE Trans. Image Process. (1)

X. Shen, C. R. Dietlein, E. Grossman, Z. Popovic, and F. G. Meyer, “Detection and segmentation of concealed objects in terahertz images,” IEEE Trans. Image Process. 17(12), 2465–2475 (2008).
[CrossRef] [PubMed]

Int. J. Comput. Vis. (1)

C. D. Haworth, Y. de Saint-Pern, D. Clark, E. Trucco, and Y. R. Petillot, “Detection and tracking of multiple metallic objects in millimeter-wave images,” Int. J. Comput. Vis. 71(2), 183–196 (2007).
[CrossRef]

J. Infrared, Millimeter Terahertz Waves (1)

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, I. Mehdi, and P. H. Siegel, “An approach for sub-second imaging of concealed objects using terahertz (THz) radar,”J. Infrared, Millimeter Terahertz Waves 30, 1297–1307 (2009).

Opt. Express (3)

D.-S. Lee, S. Yeom, J. Y. Son, and S.-H. Kim, “Automatic image segmentation for concealed object detection using the expectation-maximization algorithm,” Opt. Express 18(10), 10659–10667 (2010).
[CrossRef] [PubMed]

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Supplementary Material (4)

» Media 1: AVI (3401 KB)     
» Media 2: AVI (100 KB)     
» Media 3: AVI (100 KB)     
» Media 4: AVI (100 KB)     

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Figures (8)

Fig. 1
Fig. 1

(a) Passive MMW imaging system. (b) A block diagram of the quasi-optical configuration.

Fig. 2
Fig. 2

Overall procedure of concealed object segmentation.

Fig. 3
Fig. 3

VQ process.

Fig. 4
Fig. 4

EM algorithm.

Fig. 5
Fig. 5

(a) picture of the gun; (b) visual movie (Media 1).

Fig. 6
Fig. 6

Movies of (a) passive MMW images (Media 2), (b) full process segmentation (Media 3), (c) shortened process segmentation (Media 4).

Fig. 7
Fig. 7

GMM fitting results after (a) global segmentation, (b) local segmentation. Segmented images after (c) global segmentation stage, (d) local segmentation stage.

Fig. 8
Fig. 8

(a) Average probability of error. (b) Computational time.

Equations (3)

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p ( x j ) = k = 1 n k N ( x j | μ k , Σ k ) P ( G k ) , j = 1 , ... , n p ,
P ( G 1 | x j ) > < G 2 G 1 P ( G 2 | x j ) ,       j = 1 , ... , n p .
P e = | F S F O | + | F O F S | | I | ,

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