Affine-invariant Shape Recognition Using Grassmann Manifold

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½ 38 ½ 2 Þ Vol. 38, No. 2 2012 2 ACTA AUTOMATICA SINICA February, 2012 Ù Grassmann» å» Ç 1, 2, 3, 4 Ó¾å 5 Ä ý Kendall» èñò ó Ù Õ, ää ÁÙµ» ì»î Ì å ǑÜ. Ù Grassmann»Ò, Åå» èñ ê,  Š٠Grassmann» å» Ç ß. ß Þ» ÇÇ» à«,» Æ èñ ;, ý»» ºµý» Õ,» ½ ã. MPEG 7» ÂÄ, Kendall» Ù Procrustean Ç ß ³, Ç ß Û ; ñ Á Ç Â èä, ß å» ó Í, ñ Í Ô Î Á Ç. ³» Ç, Grassmann», å,» èñ,» DOI 10.3724/SP.J.1004.2012.00248 Affine-invariant Shape Recognition Using Grassmann Manifold 1, 2, 3 LIU Yun-Peng 1,2, 3, 4 LI Guang-Wei 5 SHI Ze-Lin 1,2, 3 Abstract Traditional Kendall shape space theory is only applied to similar transform. However, geometric transforms of the object in the imaging process should be represented by affine transform at most situations. We analyze the nonlinear geometry structure of the affine invariant shape space and propose an affine-invariant shape recognition algorithm based on Grassmann manifold geometry. Firstly, we compute the mean shape and covariance for every shape class in the train sets. Then, we construct their norm probability models on the tangent space at each mean shape. Finally, we compute the maximum likelihood class according to the measured object and prior learned shape models. We use the proposed algorithm to recognize shapes in standard shape dataset and real images. Experiment results on MPEG-7 shape dataset show that our recognition algorithm outperforms the algorithm based on Procrustean metric in traditional Kendall shape space theory. Experiment results on real images also show that the proposed algorithm exhibits higher capacity to affine transform than the Procrustean metric based algorithm and can recognize object classes with higher posterior probability. Key words Shape recognition, Grassmann manifold, affine invariant, shape space, mean shapes Ç ü ø ñ íõå Á Æ ü Æ, íµ» Ò,,» (Shape) ðê è Æ. Ù» ² á Á àå è Ò Þ Þ, ǑÜÅ Á ï Æ,» Ç Á» ǑÇ Ç ÁÂ.» þú 2010-06-11 õ 2010-10-13 Manuscript received June 11, 2010; accepted October 13, 2010 Á (60603097), Á ÁãÅ (CXJJ-65) Supported by National Natural Science Foundation of China (60603097), National Defense Innovation Foundation of Chinese Academy Sciences (CXJJ-65) Á ä þ Recommended by Associate Editor HU Zhan-Yi 1. Á Þ 110016 2. Á ½ ² Ò û 110016 3. Û äòµ ü Ç û 110016 4. Á µ 100049 5. ã Ò ã 266071 1. Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016 2. Key Laboratory of Opto-Electronic Information Processing, Chinese Academy of Sciences, Shenyang 110016 3. Key Laboratory of Image Understanding and Computer Vision, Liaoning Province, Shenyang 110016 4. Graduate University of Chinese Academy of Sciences, Beijing 100049 5. Department of Management Science and Engineering, Qingdao University, Qingdao 266071 Ç ê, ¾ ÙìÍ Áã ê ä Ù.» ðä Á Æ.» èñ ùþǒ è» [1]. 20 Î 60 ã, Æ Å ÑÅ» Ò, îæ ë õ, ³ ÙÄ ÇÇ ü èå Ö ÁË. Çð, 2010 ½ 4 IEEE Transactions on Pattern Analysis and Machine Intelligence Ñ Å» ä Òµ Ö [1], Ö» Â. Ä» ³» Ñ «ð» Ç Ã.» Ä àß, Ǒ : Ù (ù º)» Ù».,». ÁË, Ù»» èñ Ò. è Kendall [2] ǑãÄ, Á» Ä

2 : Ù Grassmann» å» Ç 249 Æ, ÁÙµ Õ,» èñǒ Õ èñ,» èñ Procrustean. ³Ù» èñ ð Kendall ð ½. Zhang [3] Á Õ» üǒôá» èñ è,» Ñ «Ä»». Huckemann [4] Kendall» èñ ½»,» Þ. Han [5] Procrustean л Õ Ç ß Á. è» ð Ù» èñò Ǒ, Ç Æ Æ ðø î, Ìàßð. Ùàó Ù ÁÙµ Õ ì», ä Ä ÁÙµ» ì»î Ì å ǑÜ [6]. è Grenander [7] ǑãÄ, Á ºù Ä»,» Ä Ǒ á»,» Ñ Þ Û ðó Â., Fletcher [8] Å ñ ǑÜ Æ Á Ã. Klassen Srivastava ø» û» èñ, ûùþèñ л [9 10], ð ùþ Ë è ð Â.» º [11] Ǒðè ê» àß, è Ä Æ ½ Ë» Æ,» Þ º. àßù Å Ù øèñ Ç àß, üå» èñ ê,. ÁË í» Ã ½Å ç,  Åè» àß: ËÆ,»±», Ë Æ Ä» [12]. í Ë [13] ÁÄ Á ñ º, î» Ä Ǒ Á.» àß Ù Grenander U» èñ. ³Ù» Ç, Ò Â, Ǒ ¾, Æ áãææµ. Đ :, º Òá Æ ÄÆ áä äî, í»ùµ³ Õ»» æ Á ë Þ, Ǒ ð ½ Þ Á ÃǑ, Å Á» Ç Â [14] ;,» ðè ê», Ù øèñ Ç àßç Ù»., Ì» èñ ê,» Ç, ÐÇ Ò ÆĐ» Ã. Kendall» èñò àó Ù Õ, å ÁÙµ», å» èñ, Grassmann» Ò» Ç ß; ß å» èñ, Ð ý» ½ Bayesian Ç. ǑÅ Ý Â ß, ó MPEG 7 ÁÇ» ñ ä Â ß Kendall» èñ Ù Procrustean Ç ß ½Å ³. ½ 1 ß µ Ã, ½ 2 å» èñ, ½ 3 Å» Ç ß, ½ 4, ð. 1 à 1.1» Ä áä Á, ó Á n Ë {z 1,z 2,,z n } Á Æ,, z i = (x i,y i ) Ä è Á, Á» è n p (p = 2) Å X Ä, ó Ǒ» Å. ÔàĐ ê», rank(x) = p. 1, 1 (a) n = 60, 1(b) n = 20, Ǒ»,  ãä» (» MPEG 7» [15] ). (a) 1» Ä Fig. 1 Shape representations Ù» X 1 = {u 1,,u n } X 2 = {v 1,,v n }, Â Ú (b) u T i = Av T i + b, i = 1,,n (1), A ð 2 2 ê Å; b ð 2 1 å, û X 1» X 2 ðå. 1.2 Bayesian Ý N»,, ½ i (i = 1,,N)» C i Ð Ǒ µ i, à«ǒ Σ i, C i N(µ i,σ i ).» C i (i = 1,,N) Î P (C i ), à ٠ù» X, Bayesian ùòđ,» Ç ÞǑ Î :

250 Þ 38 Ĉ = arg max C i P (C i X) = arg max C i P (X C i )P (C i ) = arg max C i P (X C i ) (2) Ù» èñðè ê èñ, Í Ù øèñ Ç àßµ» à «;» èñ Î Á»». 2 å» èñ 2.1 Grassmann» Grassmann»Ò ð» Ò, ³» ËÆ, Đ Þ [16 17]. Grassmann» Þ ß [18] èñç [19]. Grassmann» Ù² Òëë, Ñ ÙÇ ü Ç, ËÆíµ ²² à [20] ì» Ç» [21] Đ Ã» [22]. Ǒ Å, Grassmann»ß ù þ, ³Ù Grassmann» ³ Ǒ Þ [18, 23]. ùþ 1. Grassmann» G(k,n) ð n k Å [24]. : G(k,n) = Y Ok = {Y V : V O k } (3), Y ð n k Å, Y ãä ³, V ð k k Å. Grassmann» G(k,n) Ǒ Ä Ǒ n á åèñ R n k á èñ.» M ßð Þ è p M èñ T p M ùè Ë,, Ǒ. è p G(k,n), èñǒ T p G(k,n) = {ω ω = p g,g R (n k,k)} (4) p Ǒ p. G(k,n) ùþǒ ω = tr ( ω T ω ) (5) γ : t γ (t) Ǒ γ (0) Ǒ dγ dt (0) = ω», Æ Exp p (ω) = γ (1) Å» : Exp p (ω) = pv cos (θ) + U sin (θ) (6), UθV T = SV D (ω). Æ log p (q) = UθV T, Ô θ = arctan (S), USV T = p p T q (p T q) 1., Grassmann» (p,q)»ðǒ [23] d G (p,q) = ( k i=1 θ 2 i ) 1 2 = θ 2 (7), p q Ǒ Grassmann», p q Ñ ÆǑ θ 1,,θ k, ³Ù Æ Grassmann Ð û 2. ß 1 ÅÇ Grassmann»»ÐÄ. 2 Æ Grassmann Ð Fig. 2 Principal angles and Grassmann distance ß 1. d(p,q). Å Y 1 Y 2 ;. p q Ñ»Ð; 1) Ç Y 1 Y 2 Q 1 Q 2, : p = Y 1 = Q 1, q = Y 2 = Q 2 ; 2) Ç Q T 1 Q 2 µ: USV T = SV D (Q T 1 Q 2); 3) Ç Æ θ = cos 1 S; k 4) d(p,q) = θi 2. i=1 2.2 å» èñ ó å» èñ, Grassmann». Á» Ä, (1),» X 1 = {v 1,v 2,,v n } X 2 = {u 1,u 2,,u n } å

2 : Ù Grassmann» å» Ç 251, Ǒ u 1 1 v 1 1 [ ] u V = 2 1. 1 = v 2 1 A T 0. 1 b T 1 u n 1 v n 1 (8) R 2 èñ n å 1 = [1,1,,1] T Å R n 3 á èñ V, èñ V ðå,, å» èñ Grassmann»Ä Ǒ G(3,n). Ǒ ð è» å» Grassmann» è. ùþ. ùþ 2. C 2 n ð R 2 õ n Ë {u 1,u 2,,u n } ; å A: C 2 n C 2 n, Ãå» èñð C 2 n ³Ù A C 2 n /A,. 3, χ C 2 n, à s(χ) C 2 n/a Ǒ χ è å». Ä è» Ù è. 3 C 2 n : å» èñ Fig. 3 Orbits of C 2 n: affine invariant shape space 3 Ù Grassmann»» Ç 3.1» èñî» èñç Î, Ç». Ù» èñðê èñ, Í Ù ø èñ àßç. ùþ 3.» M x 1,x 2,,x m Karcher µ ùþǒ µ = arg min x M m d(x,x i ) 2 (9) i=1, d(x,x i ) Ǒ M»Ð [24]. Þ [24] Ç (11), àßµ» ˱, ð, ó Grassmann» µ ß 2. ß 2. Mean (x 1,x 2,,x m ). x 1,x 2,,x m G(k,n);. Karcher : µ; 1) Þ µ = x 1 ; 2) Ç A = τ m m log µ (x i ); i=1 3) A < ε, µ, à ù 4); 4) : A µ, UΣV T ; µ: µ = µv cos ( ) + U sin ( ), ù 2). Ëã ßµ, ½ 2) τ ǑĐ. Ù ù ä», ß» Karcher. Ù» èñð», Í Ç Ò Ä» è Ñ.» èñ Ä Æ Ã Æ èñ, Æ èñ T µ G ùþî. Î ÒÇ, ðè. Ë, Æ á Ðù Õ Ð (ù Ô ). Ôó üâ Ǒ» èñ Î º. Ù èñðåèñ, ÁÇ» èñ, û ½ á. x 1,x 2,,x m G(k,n), µ = Mean (x 1,,x m ). Ù èñ T µ G û {v 1,v 2,,v m }, Æà«ÅǑ = 1 m log µ (x i )log µ (x j ) T (10) i,j,» èñ N(µ, ) Î ç Ǒ Σ; 1 ϕ(x) = (2π) m 2 Σ 1 2 exp ( 12 ) log µ (x)σ 1 log µ (x) T (11), N» ½ Ë ß : ß 3. Learn Shape Model (x 1,x 2,,x m ). x 1,x 2,,x m G(k,n);.» º S, íõ» µ Æà«1) ß 2 Ç» Karcher µ; 2) Ç Æà«: = 1 log m i,j µ (x i ) log µ (x j ) T ; 3) Æ à «Σ ½ Æ µ: {eigenvectors, eigenvalues} = eig (Σ). à üâ Æ Æå» Þ, Ǒ» º ½ á, Ç. 3.2» Ç ß Ù ù è» X, Ç Ä ð

252 Þ 38 Ç X Õ. ß : ß 4. Shape likelihood(x,s,k). ý» x,» º S, k. Õ f 1) ý» x å» èñã: log µ (x); 2) Ç x Ç S Mahalanobis Ð: log µ (x) Σ 1 log µ (x) T ; 3) Ç x ³ Ù Ç S Õ : 1 (2π) m 2 Σ 1 2 exp ( 1 2 log µ (x) Σ 1 log µ (x) T). 4 Ǒ Å Ý Â ß, Á Ç MPEG 7» ä ½, ß Matlab Ý, Ò Ǒ Pentium Dualcore 2.5 GHz ËĐǑ 2 GB.  ٠Grassmann» àß Þ [2 5]» Ù Procrustean àß ½Å³., MPEG 7» ½, Ä ëô, MPEG 7» [15] À Ǒ» Ç Ä ý. Ì íõ 70 Á, Þ Áíõ 20»., íõ 3D Á», ó üâ 16 º 2D Á. 4.1 MPEG 7» 4.1.1» ÐÞ Á 20» á ü 15 Ǒ, Ë» º, à 5» Ǒý.,» ½Å,» Ç» Ç. ß 2 Ç 16 Á». Þ» Ǒ 150. 1  4, ǑÅ ï, à Å Ò Á. 4 (a) Ǒ», 4(b) Ǒ Ù Grassmann Ç, 4 (c) Ǒ Ù Procrustean Ç. 5 Å (a)» (a) Trained shapes 4»  Fig. 4 Results of mean shapes (b) Grassmann (b) Grassmann mean (c) Procrustean (c) Procrustean mean 5 Grassmann Procrustean ³ ((a) Grassmann ; (b) Procrustean ) Fig. 5 Comparison of Grassmann mean and Procrustean mean ((a) Grassmann mean; (b) Procrustean mean)

2 : Ù Grassmann» å» Ç 253 16» Grassmann Procrustean. 5(a) Ǒ Ù Grassmann Ç, 5(b) Ǒ Ù Procrustean Ç. Ð 5 Û,» 6 Procrustean» µå.» 8 É»».» 11 è 6, Ð 6 Û» 11 Procrustean» µå. Ç» «, Æ, Grassmann ³ Procrustean,». (a) Grasssmann (a) Grassmann mean (b) Procrustean (b) Procrustean mean 6 è» 11 ³ Fig.6 Comparison of zoomed shape means of Shape 11 4.1.2» Ç 1 Ç 16», ó ß 3 16» ½ Ë. Þ», ó ü 6 Æà«Å á. 1 Đ, Þ» 5 ý». ß 4 ½» Ç, û å» Ç ÃǑ ½. 1) Ç ÃǑ 2, Ù 15» ðá ü,» Þè n, 10, 10 Ç Ǒ Ç, n Ð 10 150 Þ. 7 Å Ǒ 15 î,» ß 4 Ç Î. 8(a) Å ³ Â. ÁãÄ, Ô ÁãÄ Ç. ÂÄ : á Ç, ß Ç Ç, Ä 100 î, Ç Ù ù; ³, Ù 25 î Ù Grassmann» ß Ù Procrustean ß Ç, Ù 25 î, ß Ç Ù Ù Procrustean ß. 2) Ç ÃǑ» 100. ǑÅ Ý ß å, å» Á x y Ç 7 4»» (10) (11) Î Fig. 7 Four shapes and Shapes (10-11) s posterior probabilities

254 Þ 38 (a) (a) Number of samples (b) (b) Noise 8 Ç ÃǑ Fig. 8 Classification performance versus number of samples and noise Î ² ³ Ô. ² ³Ð 40 db 17.5 db Þ, Þè ² ³, 10, 10 Ç Ǒ Ç. 9 Å 4 Î ² ³Ǒ 30 db 25 db». 8(b) Å Ç ÃǑ Â. Á ãä² ³, Ô ÁãÄ Ç. ÂÄ, á Ç ß Ç ; ² ³ Ù 30 db î, Ù Grassmann» ß Ç Û Ù Ù Procrustean ß. 10 Ç Ǒ Ç. 10 Å 4 ÁÇ«Ǒ 16 20». 9 4 λ (è½î 30 db ; è½ Î 25dB ) Fig. 9 Four shapes with adding noise (The top row is the shapes with adding 30 db noise; the bottom row is the shapes with adding 25dB noise) 3) å» Ç ÃǑ Ù Grassmann» ß Ù Á å», ǑÅ Ý ß å» å, á µ å» ½»,»» ½ Ç. ó Þ [25] àßµ á å, µ á å Á Ç«Ä å». ÁÇ«Ð 2 20, ½. Þè ÁÇ«, 10, 10 4 å»» (è½ǒ» ; è½ ÁÇ«Ǒ 16; è½ ÁÇ«Ǒ 20) Fig. 10 Four affine deformation shapes (The top row is original shapes; the middle row s transform standard deviation is 16; the bottom row s transform standard deviation is 20.) 11 (a) Åå» Ç ÃǑ Â. ÁãIJ ³, Ô ÁãÄ Ç. ÂÄ, á å, Ù Procrustean ß Ç ; Ù Grassmann» ß Ç ÃǑ. î ó Å è, á å î, å» Î ² ³Ǒ 30, ½. Þè ÁÇ«, 10, 10 Ç Ǒ Ç.  11 (b). Ð 11 Û : Î, ß Æ Ç, á å, Ù Procrustean ß Ç, Ù Grassmann» ß Ç ÚïÓ Þ. û Û Å ß.

2 : Ù Grassmann» å» Ç 255 (a)» ½ (b) Î ² ³Ǒ 30 db (a) Expeiments on original shape (b) Experiments on shapes with adding 30dB noise 11 å» Ç ÃǑ Fig. 11 Classification performance versus affine deformation 4.2 ä 4.2.1» Ç» º Ǒ½ 4.1 MPEG 7 Ë» º, ý» ÃǑ ñø Á». ÙÁÇ», Đ» Ù Ã. Ùß ñ ä Á à ÇĐ àß ½ Đ. Ù ñ ä Á Á èåò àß ½ Đ. Á ð, ä Á Đ ð ËÆ. Ù 12 (a) Ò ß ñ ä, ó àß Đ Á:, ä I I, à I x y w = I x + I y ; Ð I w 3 ÁÇ«ǑÃ Đ I w, ø 12 (b) Æä, û Æä ø Á». Ù 12 (a) è ñ ä, ó Þ [26] àß Đ Á, û ø Á». Ç,» Ǒ 100. 12 (c) Á» Î Å Ç Ç. Ð 12 (a) Û : 1) Á Ù Ë ñ, Á» Ù» Ë ÌðÙµÅÃ, å Õ è, Çð Á: É ; Á Õ, ß» Å Á Ç; 2) Ù ½ 3 Á,», ß» Å Á Ç; 3) Ù½ 4 Á à, Ù Á Đ, ø Á» ½ 3 Á ð Õ, ß» Å Á Ç. ÂÄ ß Í ñ Ô Î Á Ç. 4.2.2 å Ý ä Æ, ÂĐ å», è ð å» Õ Ã. ǑÅ Ý ß ñ Á» å,» º Á» ý» Ð Ë ñ ø. Æ è, üâ Ǒ Á Æ, è. ä, ÐÐ Á 1 â 6 â Þ, î ä Í üæ ä. Þ Ðü 100 ä, á ü 80 Ǒ, àß 20 Ǒý, Ç Â. Ô» Ù Đ Ä ½ 4.1. 20 ý» Î Ǒ ù Ç Â, Î, Ç Â. 13 Å ä. Ç Ù Grassmann» ß Ù Procrustean ß ø ä ½, Ç» Ǒ 100. 14 Šл ³, Ð 14 Û, 4 â 5 â, Procrustean Đ Ý, 6 â Procrustean Đ Ý, Procrustean Á». 15 Å Ð Ç Â. Ð 15 Û á ÐÇ, Ù Grassmann» Ç ß Î Ç, ½ 6 â Ù ù. Ù ßð å Â, ÝÅ Ð Æ, Ã Þ å.

256 Þ 38 Fig. 13 12 ñä Â (4 Á ÇǑ 8, 10, 4, 12) Fig. 12 Results of real scence images (The correct classes are: 8, 10, 4, 12) 13 Ë ñ üæä The different view real images from near to far Ù Procrustean ßð Õ Â, Ð 13 Û, Ð 1 â 4 â, Ð Á Æ, Ã, Procrustean ß Î, 0.45. ½ 5 â½ 6 â, Ð Á, Ã å, èǒ Õ,, Î Ç, 6 â à 0.6057. Kendall» èñò Õ Á», Ë ñ ä Á», ð Ò.

2 : Ù Grassmann» å» Ç 257 Ð Æ ÂÛ, Ù Grassmann» Ç ß Ù Procrustean ß Û. Ǒ Å å Á» Ò. 14 л Fig. 14 Mean shapes of different distance piers æ», ðä ÅÝ» èñò, Ǒ» Ç Â Å Ò í, Ǒ ñ Á ÇÂ Å Ò ß.» èñ Î ºî, ð» Æ èñ Ô, Á» Í Þ, Æ, Ù» Þ ìõ, ÌĐ óô ù Bingham [27] ;, Ä ß å, íµ½ Þ ì»» Ç Ã;, Ý ß îþ ÒǑðèá. Å ðé àå. References Fig. 15 5 15 Ð Ç Â Recognition results of piers at different distances å» èñ ê, Grassmann» Ò» Ç, å» èñùþ à«, Ð ÐÇ Ò ÆĐ» Ç Ã. ÂÄ, Â Ç ß³ Ù Kendall» èñò Ç ß Í. øèñ ß 1 Srivastava A, Damon J N, Dryden I L, Jermyn I H. Guest editors introduction to the special section on shape analysis and its applications in image understanding. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2010, 32(4): 577 578 2 Kendall D G. Shape manifolds, procrustean metrics and complex projective spaces. Bulletin of London Mathematical Society, 1984, 16(2): 81 121 3 Zhang J, Zhang X, Krim H, Walter G G. Object representation and recognition in shape spaces. Pattern Recognition, 2003, 36(5): 1143 1154 4 Huckemann S, Hotz T, Munk A. Intrinsic MANOVA for Riemannian manifolds with an application to Kendall s space of planar shapes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2010, 32(4): 593 603 5 Han Y X, Wang B, Idesawa M, Shimai H. Recognition of multiple configurations of objects with limited data. Pattern Recognition, 2010, 43(4): 1467 1475

258 Þ 38 6 Huttenlocher D P, Ullman S. Recognizing solid objects by alignment with an image. International Journal of Computer Vision, 1990, 5(2): 195 212 7 Grenander U, Miller M I. Pattern Theory: from Representation to Inference. New York: Oxford University Press, 2007 8 Fletcher P T, Whitaker T R. Riemannian metrics on the space of solid shapes. In: Proceedings of the International Workshop on Mathematical Foundations of Computational Anatomy. Copenhagen, Denmark: MICCAI, 2006. 1 11 9 Klassen E, Srivastava A, Mio W, Joshi S. Analysis of planar shapes using geodesic paths on shape spaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2004, 26(3): 372 383 10 Srivastava A, Joshi S, Mio W, Liu X W. Statistical shape analysis: clustering, learning and testing. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2005, 27(4): 590 602 11 Cootes T F, Taylor C J, Cooper D H, Graham J. Active shape models: their training and application. Computer Vision and Image Understanding, 1995, 61(1): 38 59 12 Wang B, Chen Y Q. An invariant shape representation: interior angle chain. International Journal of Pattern Recognition and Artificial Intelligence, 2007, 21(3): 543 559 13 Chen Xiao-Chun, Ye Mao-Dong, Ni Chen-Min. A method for shape recognition. Pattern Recognition and Artificial Intelligence, 2006, 19(6): 758 763 (í Ë, ãåý, ç. è» Ç àß. Ç ì Í, 2006, 19(6): 758 763) 14 Mumford D. Mathematic theories of shape: do they model perception? In: Proceedings of the Conference on Geometric Methods in Computer Vision. San Diego, USA: SPIE, 1991. 2 10 15 Latecki L J, Lakimper R, Eckhardt U. Shape descriptors for non-rigid shapes with a single closed contour. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. Hilton Head Island, USA: IEEE, 2000. 424 429 16 Spivak M. A Comprehensive Introduction to Differential Geometry. Berlin: Berkeley, 1979 17 Berger M. A Panoramic View of Riemannian Geometry. Berlin: Springer, 2003 18 Edelman A, Arias T A, Smith S T. The geometry of algorithms with orthogonality constraints. SIAM Journal on Matrix Analysis and Applications, 1999, 20(2): 303 353 19 Lin D, Yan S, Tang X. Pursuing informative projection on Grassmann manifold. In: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition. New York, USA: IEEE, 2006. 1727 1734 20 Zhang L, Tse D N. Communication on the Grassmann manifold: a geometric approach to the noncoherent multipleantenna channel. IEEE Transactions on Information Theory, 2002, 48(2): 359 383 21 Amsallem D, Farhat C. An interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA Journal, 2008, 46(7): 1803 1813 22 Subbarao R, Meer P. Nonlinear mean shift over Riemannian manifolds. International Journal of Computer Vision, 2009, 84(1): 1 20 23 Absil P A, Mahoney R, Sepulchre R. Riemannian geometry of Grassmann manifolds with a view on algorithmic computation. Arta Applicandae Mathematicae, 2004, 80(2): 199 220 24 Fletcher P T, Lu C, Pizer S M, Joshi S. Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Transactions on Medical Imaging, 2004, 23(8): 995 1005 25 Baker S, Matthews I. Lucas-Kanade 20 years on: a unifying framework. International Journal of Computer Vision, 2004, 56(3): 221 255 26 Marszalek M, Schmid C. Accurate object localization with shape masks. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. Minnesota, USA: IEEE, 2007. 1 8 27 Dryden I L, Mardia K V. Statistical Shape Analysis. New York: John Wiley and Sons, 1998 Á Þ µ. àåǒ Á Ð Ç. ². E-mail: ypliu@sia.cn (LIU Yun-Peng Ph. D. candidate at Shenyang Institute of Automation, Chinese Academy of Sciences. His research interest covers object tracking and recognition. Corresponding author of this paper.) Ó¾å ã ã. àå Ǒ Á Ð Ç, åëë. E-mail: liguangweispacetime@gmail.com (LI Guang-Wei Lecturer at Qingdao University. His research interest covers object tracking and recognition, and robust control.) Ä Á Þ Å. àåǒä Ò, Ç, ìíëë. E-mail: zlshi@sia.cn (SHI Ze-Lin Professor at Shenyang Institute of Automation, Chinese Academy of Sciences. His research interest covers image processing, pattern recognition, and intelligent control.)

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