(2001) 49 1 23 42 2000 10 16 2001 4 23 NTT * 1. 1.1 1998 * 104 0033 1 21 2 7F
24 49 1 2001 1999 70 91 MIT M. Turk Recognition Using Eigenface (Turk and Pentland (1991)). 1998 IC 1 CPU (Jain and Waller (1978), Raudys (1981)) 1997 Chellapa et al. 1995 Pentland and T. Choudbury 2000
25 1.2 CCD CRT x Gabor filter x x 2 2 NTTdata NTT 79 10 10 16 12
26 49 1 2001 1. NTTdata 2. UMIST 1 UMIST (Graham and Allinson (1998)). 20 40 60 15 10 3 50 15 15 2
27 2. 2.1 (Jain et al. (2000)) M. Turk Eigenface Turk M X all = { x 1,..., x i,..., x M } C all = 1 M µ all = 1 M MX MX j=1 MX x i ( µ x i )( µ x j ) T (2.1) C all v = λ v λ j v j j all m X k = { x k 1,..., x k i,..., x k m} d η = { v 1,..., v d } d { ξ k 1,..., ξ k m} µ k k ID x ξ µ k (2.2) D 2 = µ k ξ 2
28 49 1 2001 1. (NTTdata 1) k 1 10 1) 2) (Belhumer and Kriegman (1996)) 3 10 20 2.2 k 90 SVM (Support Vector Machine) Duda et al. 2001 Watanabe (Watanabe and Pakvasa (1973)).
29 1986. 1996 1989 Watanabe CLAFIC CLAFIC r X k = { x k 1,..., x k i,..., x k r} C k = 1 r µ k = 1 r rx rx rx j=1 x k i x k i x kt j (2.3) C k v k = λ k v k λ k j v j k q k ID j CLAFIC s (2.4) s = 1 x qx ( v i x) q UMIST 2
30 49 1 2001 2. (UMIST) 1 100 3. 3.1 1997 1985 v i CLAFIC s s = 1 x qx v i x q v j (3.1) X ij = qx l=1 ( v i v l )( v l v j ) (3.2) λ x = X ij x 1985
31 3 3.2 Nayar 1994 1995 Gnanadesikan 1977, Saito and Kariya 1988, (Diamantaras and Kung (1996), Oja (1982)), EM (Frey et al. (1998)) Turk Eigenface Bar et al. (1998) Moghaddam (1999) Schölkopf et al. (1998) Schölkopf Kernel PCA 1999 1 Schölkopf et al. (1998) SVM SVM
32 49 1 2001 1978 k m { x k 1,..., x k j,..., x k m} x (3.3) s k = k( x k i, x) s k Mercer (3.4) k(x, y) = X ψ i (x)ψ i (y) λ i 1953 (3.5) k(x, y) = X ψ i (x)ψ i (y) λ i = X ψ i (x) ψ i (y) =(Ψ(x) Ψ(y)) λi λi Schölkopf (3.6) Ψ : R N F, x X F (3.7) C = 1 m j=1 (Ψ( x j )Ψ( x j ) T ) F Ψ( x j )Ψ( x j ) T F (3.8) X Ψ( x j )(Ψ( x j ) X) F L 2 (3.9) > (Ψ( x) Ψ( x)) F V F\{0} C (3.10) λv = CV R N (3.11) λ(ψ( x k ) V )=(Ψ( x k ) CV ) α (3.12) V = α i Ψ( x i )
33 (3.11)(3.12) (3.13) λ m m α i (Ψ( x k ) Ψ( x i )) = 1 m α i ψψ( x k ) (3.14) K ij =(Ψ( x i ) Ψ( x j )) (3.13) (3.15) λ α i K ki = 1 m = 1 m α i ψψ( x k ) j=1 j=1 j=1 α i K kj K ji Ψ( x j )(Ψ( x j ) Ψ( x i )) Ψ( x j )K ji! α i (3.16) mλαk = αk 2 K 1 (3.17) mλα = αk α α x V (3.18) V Ψ( x) = 1 λ α i (Ψ( x i ) Ψ( x)) Ψ( ) Ψ( x) Ψ( y ) Ψ( ) (3.5) Ψ( x) Ψ( y ) k( x, y ) (3.19) V Ψ( x) = 1 λ α i k( x i, x) V k(x i,x j ) x i x j k( x i, x j )=r( x i x j ) x i x j ij = p 2(1 r( x i x j )) (Williams (2001)) Williams kernel MDS kernel MultiDimensional Scaling;!
34 49 1 2001 (1999) (1999) V λ x i x W ν V W V W (V W ) (V W ) F (3.12) V W (3.20) W = (3.21) V W = = Xm j=1 (3.5) (3.22) V W = Xm j=1 α jψ( x j) α i Ψ( x i ) Xm j=1 α jψ( x j) α i α j(ψ( x i ) Ψ( x j)) Xm j=1 α i α jk( x i, x j) (3.1)(3.2) (3.22) 1999 2001 SVM (Radial Basis Function; RBF) x y 2 (3.23) k( x, y ) = exp, σ 2 (3.24) k( x, y )=(1+ x y ) m, (3.25) k( x, y )=( x y ) m
35 3. σ,m UMIST (1999) (2001) 3 UMIST 1 2 2 2 4 5 2 4 3 4 1 100 5 outlier 1 1
36 49 1 2001 4. 5. 5
37 3.3 1. 2. 3 3. 1. Adini et al. 1997. 2 3 3 D. Marr 3 1999 Georghiades Illumination cone (Georghiades et al. (1998)). 3 Lambert (3.26) I = ρ( n l ) I l n Shashua (Shashua (1992)) Belhumeur and Kriegman (1996) Lambert I(4) 3
38 49 1 2001 6. 3 I(1),I(2),I(3) (3.27) I(4) = max ψ 3 X j=1 a(j)i(j), 0 6 a(j) 3 Î(1), Î(2), Î(3) Belhumeur and Kriegman (1998) Illumination cone Georghiades Illumination cone 3 3 Î(1), Î(2), Î(3) Illumination cone 3 k m I k (1),...,I k (m) 3 Î k (1), Î k (2), Î k (3) I Illumination cone (3.28) (s k ) 2 = I 2 3X j (Îk (j) I) 2 k Georghiades!
39 100 Illumination cone 3 2 3 Tomasi and Kanade 1992 1999 1998 3 F P P F 2P W W = R S R 3 R 2P 3 Illumination cone 3 Illumination cone Illumination cone (1998) Maki et al. (1998) Geotensity 4. UMIST Allinson NTT NTT
40 49 1 2001 Adini, Y., Moses, Y. and Ullman, S. (1997). Face recognition: The problem of compensating for changes in illumination direction, IEEE Trans. Pattern Analysis and Machine Intelligence, 19, 721 732. M. A., E. M., L. I. (1978). (1997). J80 D II(8), 2031 2046. Bar, S. D., Edelman, S., Howell, A. J. and Buxton, H. (1998). A similarity based method for generalization of face recognition over pose and expression, Proceedings of IEEE Automatic Recognition of Face and Gesture 98, 118 123. Belhumeur, P. B. and Kriegman, D. (1996). What is the set of images of an object under all possible illumination conditions?, Proceedings of International Conference of Computer Vision, 1053 1059. Belhumeur, P. B. and Kriegman, D. (1998). What is the set of images of an object under all possible illumination conditions?, International Journal of Computer Vision, 28(3), 245 260. Belhumeur, P. B., Hespanha, J. P. and Kriegman, D. J. (1997). Eigenface vs. Fisharfaces: Recognition using class specific linear projection, IEEE Trans. Pattern Analysis and Machine Intelligence, 19, 711 720. Chellapa, R., Wilson, C. J. and Sirohey, S. (1995). Human and machine recognition of face: A survey, Proceedings of IEEE, 83(5), 704 740. Diamantaras, K. I. and Kung, S. Y. (1996). Principal Component Neural Networks, Wiley, New York. Duda, R. O., Hart, P. E. and Stork, D. G. (2001). Pattern Classification, 2nd ed., Wiley, New York. Frey, B. J., Colmenarez, A. and Huang, T. S. (1998). Mixtures of local linear subspaces for face recognition, Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, 32 37. Georghiades, A. S., Kriegman, D. J. and Belhumeur, P. N. (1998). Illumination cones for recognition under variable lighting: Faces, Proceedings of Computer Vision and Pattern Recognition, 52 58. Gnanadesikan, R. (1977). Methods for Statistical Data Analysis of Multivariate Observations, Wiley, New York. 1979. Graham, D. B. and Allinson, N. S. (1998). Characterizing virtual eigensignatures for general purpose face recognition, Face Recognition: From Theory to Applications (eds. H. Wechsler et al.), p. 446, Springer, Berlin. (1998). PRMU98 134. (1989).. (1998). Jain, A. K. and Waller, W. G. (1978). On the optimal number of features in the classification multivariate Gaussian data, Pattern Recognition, 10, 365 374. Jain, A. K., Duin, R. P. W. and Mao, J. (2000). Statistical pattern recognition: A review, IEEE Trans. Pattern Analysis and Machine Intelligence, 22, 4 37. (1999). (1953)..
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42 Proceedings of the Institute of Statistical Mathematics Vol. 49, No. 1, 23 42 (2001) Principal Component Analysis in Pattern Recognition From the Viewpoint of Facial Image Recognition Hitoshi Sakano (NTT Data Corporation) In this article, we introduce How to use the principal component analysis in facial image recognition. We also introduce some improvement in the fields. First, we describe the role of principal component analysis in image recognition technology. And we point out some difficulties in facial image recognition technology, for example image change caused by illumination change, nonlinear distribution caused by head pose change. Finally, we introduce some improvement of principal component analysis and how to solve the problems. Key words: Pattern recognition, computer vision, principal component analysis, facial image recognition.