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1 (2001) NTT * * F

2 MIT M. Turk Recognition Using Eigenface (Turk and Pentland (1991)) IC 1 CPU (Jain and Waller (1978), Raudys (1981)) 1997 Chellapa et al Pentland and T. Choudbury 2000

3 CCD CRT x Gabor filter x x 2 2 NTTdata NTT

4 NTTdata 2. UMIST 1 UMIST (Graham and Allinson (1998))

5 (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

6 (NTTdata 1) k ) 2) (Belhumer and Kriegman (1996)) k 90 SVM (Support Vector Machine) Duda et al Watanabe (Watanabe and Pakvasa (1973)).

7 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

8 (UMIST) 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

9 Nayar 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 Schölkopf et al. (1998) SVM SVM

10 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 )

11 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;!

12 (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) 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

13 35 3. σ,m UMIST (1999) (2001) 3 UMIST outlier 1 1

14

15 Adini et al D. Marr 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

16 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!

17 Illumination cone Tomasi and Kanade 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

18 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, M. A., E. M., L. I. (1978). (1997). J80 D II(8), 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, 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, 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), 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, Chellapa, R., Wilson, C. J. and Sirohey, S. (1995). Human and machine recognition of face: A survey, Proceedings of IEEE, 83(5), 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, 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, Gnanadesikan, R. (1977). Methods for Statistical Data Analysis of Multivariate Observations, Wiley, New York 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). PRMU (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, Jain, A. K., Duin, R. P. W. and Mao, J. (2000). Statistical pattern recognition: A review, IEEE Trans. Pattern Analysis and Machine Intelligence, 22, (1999). (1953)..

19 41 (1996). PRU (1999). (D II), J82 D II(4), (1985). (D), J68 D(3), Maki, A., Watanabe, M. and Wiles, C. (1998). Geotensity: Combining motion and lighting for 3D surface reconstruction, Proceedings of International Conference of Computer Vision 98, (1998). Moghaddam, B. (1999). Principal manifolds and Bayesian subspaces for visual recognition, Proceedings of International Conference of Computer Vision, (1995). CVCV WG (VI) CV97 9, Nayar, S. K. (1994). 2 3 J77 D II(11), (1998). PRMU Oja, E. (1982). A simplified neuron model as a principal component analyser, J. Math. Biol., 15, (1986).. Pentland, A. and Choudbury, T. (2000). Face recognition for smart environments, IEEE Computer, 33, p. 50. Raudys, S. J. (1981). Determination of optimal dimensionality in statistical pattern recognition, Pattern Recognition, 11, Saito, T. and Kariya, T. (1988). A generalization of principal component analysis, J. Japan Statist. Soc., 18(2), (1999). PRMU (1999). PRMU (2001). J84 D II(8) Schölkopf, B., Smola, A. and Müller, K. R. (1998). Nonlinear component analysis as a kernel eigenvalue problem, Neural Computation, 10, Shashua, Ammon (1992). Illumination and view position in 3D visual recognition, Advances in Neural Information Processing, 4, Tomasi, C. and Kanade, T. (1992). Shape and motion from image stream under orthography: A factorization method, International Journal of Computer Vision, 9(2), (1999). (D II), J82 D II(4), Turk, M. and Pentland, A. (1991). Face recognition using eigenfaces, Proceedings of Computer Vision and Pattern Recognition, Watanabe, S. and Pakvasa, N. (1973). Subspace method of pattern recognition, Proceedings of 1st International Joint Conference of Pattern Recognition, Williams, C. K. I. (2001). On a connection between kernel PCA and metric multidimensional scaling, Advances in Neural Information Processing Systems 13 (eds. T. K. Leen, T. G. Diettrich and V. Tresp), MIT Press, Cambridge, Massachusetts (to appear). (1997). PRMU97 50.

20 42 Proceedings of the Institute of Statistical Mathematics Vol. 49, No. 1, (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.

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