Classifying Video with Kernel Dynamic Textures

Transcription

1 Appears i IEEE Cof. o Computer Visio ad Patter Recogitio, Mieapolis, 27. Classifyig Video with Kerel Dyamic Textures Atoi B. Cha ad Nuo Vascocelos Departmet of Electrical ad Computer Egieerig Uiversity of Califoria, Sa Diego abcha@ucsd.edu, uo@ece.ucsd.edu Abstract The dyamic texture is a stochastic video model that treats the video as a sample from a liear dyamical system. The simple model has bee show to be surprisigly useful i domais such as video sythesis, video segmetatio, ad video classificatio. However, oe major disadvatage of the dyamic texture is that it ca oly model video where the motio is smooth, i.e. video textures where the pixel values chage smoothly. I this work, we propose a extesio of the dyamic texture to address this issue. Istead of learig a liear observatio fuctio with PCA, we lear a o-liear observatio fuctio usig kerel- PCA. The resultig kerel dyamic texture is capable of modelig a wider rage of video motio, such as chaotic motio (e.g. turbulet water) or camera motio (e.g. paig). We derive the ecessary steps to compute the Marti distace betwee kerel dyamic textures, ad the validate the ew model through classificatio experimets o video cotaiig camera motio.. Itroductio The dyamic texture [] is a geerative stochastic model of video that treats the video as a sample from a liear dyamical system. Although simple, the model has bee show to be surprisigly useful i domais such as video sythesis [, 2], video classificatio [3, 4, 5], ad video segmetatio [6, 7, 8, 2]. Despite these umerous successes, oe major disadvatage of the dyamic texture is that it ca oly model video where the motio is smooth, i.e. video textures where the pixel values chage smoothly. This limitatio stems from the liear assumptios of the model: specifically, ) the liear state-trasitio fuctio, which models the evolutio of the hidde state-space variables over time; ad 2) the liear observatio fuctio, which maps the state-space variables ito observatios. As a result, the dyamic texture caot model more complex motio, such as chaotic motio (e.g. turbulet water) or camera motio (e.g. paig, zoomig, ad rotatios). To some extet, the smoothess limitatio of the dyamic texture has bee addressed i the literature by modifyig the liear assumptios of the dyamic texture model. For example, [9] keeps the liear observatio fuctio, while modelig the state-trasitios with a closed-loop dyamic system. I cotrast, [, ] utilize a o-liear observatio fuctio, modeled as a mixture of liear subspaces, while keepig the stadard liear state-trasitios. Similarly i [2], differet views of a video texture are represeted by a o-liear observatio fuctio that models the video texture maifold from differet camera viewpoits. Fially, [7] treats the observatio fuctio as a piece-wise liear fuctio that chages over time, but is ot a geerative model. I this paper, we improve the modelig capability of the dyamic texture by usig a o-liear observatio fuctio, while maitaiig the liear state trasitios. I particular, istead of usig PCA to lear a liear observatio fuctio, as with the stadard dyamic texture, we use kerel PCA to lear a o-liear observatio fuctio. The resultig kerel dyamic texture is capable of modelig a wider rage of video motio. The cotributios of this paper are three-fold. First, we itroduce the kerel dyamic texture ad describe a simple algorithm for learig the parameters. Secod, we show how to compute the Marti distace betwee kerel dyamic textures, ad hece itroduce a similarity measure for the ew model. Third, we build a video classifier based o the kerel dyamic texture ad the Marti distace, ad evaluate the efficacy of the model through a classificatio experimet o video cotaiig camera motio. We begi the paper with a brief review of kerel PCA, followed by each of the three cotributios listed above. 2. Kerel PCA Kerel PCA [3] is the kerelized versio of stadard PCA [4]. With stadard PCA, the data is projected oto the liear subspace (liear pricipal compoets) that best captures the variability of the data. I cotrast, kerel PCA (KPCA) projects the data oto o-liear fuctios i the iput-space. These o-liear pricipal compoets are de-

2 fied by the kerel fuctio, but are ever explicitly computed. A alterative iterpretatio is that kerel PCA first applies a o-liear feature trasformatio to the data, ad the performs stadard PCA i the feature-space. Give a traiig data set of N poits Y = [y,..., y N ] with y i R m ad a kerel fuctio k(y, y 2 ) with associated feature trasformatio φ(y), i.e. k(y, y 2 ) = φ(y ), φ(y 2 ), the c-th kerel pricipal compoet i the feature-space has the form [3] (assumig the data has zeromea i the feature-space): v c = N α i,c φ(y i ) () i= The KPCA weight vector α c = [α,c,...,α N,c ] T is give by α c = λc v c, where λ c ad v c are the c-th largest eigevalue ad eigevector of the kerel matrix K, which has etries [K] i,j = k(y i, y j ). Fially, the KPCA coefficiets X = [x,, x N ] of the traiig set Y are give by X = α T K, where α = [α,, α ] is the KPCA weight matrix, ad is the umber of pricipal compoets. Several methods ca be used to recostruct the iput vector from the KPCA coefficiets, e.g. miimum-orm recostructio [5, 6], or costraied-distace recostructio [7]. Fially, i the geeral case, the KPCA equatios ca be exteded to ceter the data i the feature-space if it is ot already zero-mea (see [8] for details). 3. Kerel Dyamic Textures I this sectio, we itroduce the kerel dyamic texture. We begi by briefly reviewig the stadard dyamic texture, followed by its extesio to the kerel dyamic texture. 3.. Dyamic texture A dyamic texture [] is a geerative model for video, which treats the video as a sample from a liear dyamical system. The model, show i Figure, separates the visual compoet ad the uderlyig dyamics ito two stochastic processes. The dyamics of the video are represeted as a time-evolvig state process x t R, ad the appearace of the frame y t R m is a liear fuctio of the curret state vector with some observatio oise. Formally, the system equatios are { xt = Ax t + v t y t = Cx t + w t (2) where A R is the state-trasitio matrix, C R m is the observatio matrix, ad x R is the iitial coditio. The state ad observatio oise are give by v t iid N(, Q,) ad w t iid N(, ri m ), respectively. x x 2 x 3 x 4 y y 2 y 3 y 4... Figure. Graphical model of the dyamic texture. Algorithm Learig a kerel dyamic texture Iput: Video sequece [y,..., y N ], state space dimesio, kerel fuctio k(y, y 2 ). Compute the mea: ȳ = N N i=t y t. Subtract the mea: y t y t ȳ, t. Compute the (cetered) kerel matrix [K] i,j = k(y i, y j ) Compute KPCA weights α from K. [ˆx xˆ N ] = α T K  = [ˆx 2 ˆx N ][ˆx ˆx N ] ˆv t = ˆx t ˆx t, t ˆQ = N N t= ˆv tˆv t T ŷ t = C(ˆx t ), t, (e.g. miimum-orm recostructio). ˆr = N mn t= y t ŷ t 2 Output: α, Â, ˆQ, ˆr, ȳ Whe the parameters of the model are leared usig the method of [], the colums of C are the pricipal compoets of the video frames (i time), ad the state vector is a set of PCA coefficiets for the video frame, which evolve accordig to a Gauss-Markov process Kerel Dyamic Textures Cosider the extesio of the stadard dyamic texture where the observatio matrix C is replaced by a o-liear fuctio C(x t ) of the curret state x t, { xt = Ax t + v t y t = C(x t ) + w t (3) I geeral, learig the o-liear observatio fuctio ca be difficult sice the state variables are ukow. As a alterative, the iverse of the observatio fuctio, i.e. the fuctio D(y) : R m R that maps observatios to the state-space, ca be leared with kerel PCA. The estimates of the state variables are the the KPCA coefficiets, ad the state-space parameters ca be estimated with the leastsquares method of []. The learig algorithm is summarized i Algorithm. We call a o-liear dyamic system, leared i this maer, a kerel dyamic texture because it uses kerel PCA to lear the state-space variables, rather tha PCA as with the stadard dyamic texture. Ideed whe the kerel fuctio is the liear kerel, the learig algorithm reduces to that of []. The kerel dyamic texture has two iterpretatios: ) kerel PCA lears the o-liear observatio fuctio 2

3 Origial DT KDT sie triagle ramp 5 5 t 5 5 t 5 5 t Figure 2. Sythesis examples: (left) The origial time-series (sie wave, triagle wave, or periodic ramp); ad a radom sample geerated from: (middle) the dyamic texture; ad (right) the kerel dyamic texture, leared from the sigal. The two dimesios of the sigal are show i differet colors. C(x), which cotais o-liear pricipal compoets; or 2) kerel PCA first trasforms the data with the featuretrasformatio φ(y) iduced by the kerel fuctio, ad the a stadard dyamic texture is leared i the featurespace. This feature-space iterpretatio will prove useful i Sectio 4, where we compute the Marti distace betwee kerel dyamic textures Sythetic examples I this sectio we show the expressive power of the kerel dyamic texture o some simple sythetic time-series. Figure 2 (left) shows three two-dimesioal time-series: ) a sie wave, 2) a triagle wave, ad 3) a periodic ramp wave. Each time-series has legth 8, ad cotais two periods of the waveform. The two-dimesioal time-series was projected liearly ito a 24-dimesioal space, ad Gaussia i.i.d. oise (σ =.) was added. Note that the triagle wave ad periodic ramp are ot smooth sigals i the sese that the former has a discotiuity i the first derivative, while the latter has a jump-discotiuity i the sigal. A dyamic texture ad a kerel dyamic texture were leared from the 24-dimesioal time-series, with statespace dimesio = 8. Next, a radom sample of legth 6 was geerated from the two models, ad the 24- dimesioal sigal was liearly projected back ito two dimesios for visualizatio. Figure 2 shows the sythesis results for each time-series. The kerel dyamic texture is able to model all three time-series well, icludig the more difficult triagle ad ramp waves. This is i cotrast to the dyamic texture, which ca oly represet the sie wave. For the triagle wave, the dyamic texture fails to capture the sharp peaks of the triagles, ad the sigal begis to degrade after t = 8. The dyamic texture does ot capture The cetered RBF kerel with width computed from () was used. ay discerible sigal from the ramp wave. The results from these simple experimets idicate that the kerel dyamic texture is better at modelig arbitrary time-series. I particular, the kerel dyamic texture ca model both discotiuities i the first derivative of the sigal (e.g. the triagle wave), ad jump discotiuities i the sigal (e.g. the periodic ramp). These types of discotiuities occur frequetly i video textures cotaiig chaotic motio (e.g. turbulet water), or i texture udergoig camera motio (e.g. paig across a sharp edge). While the applicatio of the kerel dyamic texture to video sythesis is certaily iterestig ad a directio of future work, i the remaider of this paper we will focus oly o utilizig the model for classificatio of video textures Other related works The kerel dyamic texture is related to o-liear dyamical systems, where both the state trasitios ad the observatio fuctios are o-liear fuctios. I [9], the EM algorithm ad the exteded Kalma filter are used to lear the parameters of a o-liear dyamical system. I [2], the oliear mappigs are modeled as multi-layer perceptro etworks, ad the system is leared usig a Bayesia esemble method. These methods ca be computatioally itesive because of the may degrees of freedom associated with both o-liear state-trasitios ad o-liear observatio fuctios. I cotrast, the kerel dyamic texture is a model where the state-trasitio is liear ad the observatio fuctio is o-liear. The kerel dyamic texture also has coectios to dimesioality reductio; several maifold-embeddig algorithms (e.g. ISOMAP, LLE) ca be cast as kerel PCA with kerels specific to each algorithm [2]. Fially, the kerel dyamic texture is similar to [22, 23], which lears appear- 3

4 ace maifolds of video that are costraied by a Gauss- Markov state-space with kow parameters A ad Q. 4. Marti distace for kerel dyamic textures Previous work [3] i video classificatio used the Marti distace as the similarity distace betwee dyamic texture models. The Marti distace [24] is based o the pricipal agles betwee the subspaces of the exteded observability matrices of the two textures [25]. Formally, let Θ a = {C a, A a } ad Θ b = {C b, A b } be the parameters of two dyamic textures. The Marti distace is defied as d 2 (Θ a, Θ b ) = log cos 2 θ i (4) i= where θ i is the i-th pricipal agle betwee the exteded observability matrices O a ad O b, defied as O a = [ C T a A T a C T a (A T a ) C T a ]T, ad similarly for O b. It is show i [25] that the pricipal agles ca be computed by solvig the followig geeralized eigevalue problem: [ O ab (O ab ) T ] [ x y ] [ ][ Oaa x = λ O bb y subject to x T O aa x = ad y T O bb y =, where O ab = (O a ) T O b = (A t a )T Ca T C ba t b (6) t= ad similarly for O aa ad O bb. The first largest eigevalues are the cosies of the pricipal agles, ad hece d 2 (Θ a, Θ b ) = 2 log λ i (7) i= The Marti distace for kerel dyamic textures ca be computed by usig the iterpretatio that the kerel dyamic texture is a stadard dyamic texture leared i the feature-space of the kerel. Hece, i the matrix F = Ca TC b, the ier-products betwee the pricipal compoets ca be replaced with the ier-products betwee the kerel pricipal compoets i the feature-space. However, this ca oly be doe whe the two kerels iduce the same ier-product i the same feature-space. Cosider two data sets {yi a}na i= ad {yb i }N b i=, ad two kerel fuctios k a ad k b with feature trasformatios φ(y) ad ψ(y), i.e. k a (y, y 2 ) = φ(y ), φ(y 2 ) ad k b (y, y 2 ) = ψ(y ), ψ(y 2 ), which share the same ierproduct ad feature-spaces. Ruig KPCA o each of the data-sets with their kerels yields the KPCA weight matrices α ad β, respectively. The c-th ad d-th KPCA compoets i each of the feature-spaces are give by, i= i= ] (5) N a u c = α i,c φ(yi a ), v N b d = β i,d ψ(yi b ). (8) Figure 3. Examples from the UCLA-pa video texture database. The ier-product betwee these two KPCA compoets is u c, v d = Na N b α i,c φ(yi a ), β i,d ψ(yi b ) i= i= (9) = α T c Gβ d () where G is the matrix with etries [G] i,j = g(yi a, yb j ), ad g(y, y 2 ) = φ(y ), ψ(y 2 ). The fuctio g is the ierproduct i the feature-space betwee the two data-poits, trasformed by two differet fuctios, φ(y) ad ψ(y). For two Gaussia kerels with badwidth parameters σa 2 ad σb 2, it ca be show that g(y, y 2 ) = exp( 2 σ a y σ b y 2 2 ) (see [8] for details). Fially, the ier product matrix betwee all the KPCA compoets is F = α T Gβ. 5. Experimetal evaluatio I this sectio we evaluate the efficacy of the kerel dyamic texture for classificatio of video textures udergoig camera motio. 5.. Databases The UCLA dyamic texture database [3, 4] cotais 5 classes of various video textures, icludig boilig water, foutais, fire, waterfalls, ad plats ad flowers swayig i the wid. Each class cotais four grayscale sequeces with 75 frames of 6 pixels. Each sequece was clipped to a widow that cotaied the represetative motio. A secod database cotaiig paig video textures was built from the origial UCLA video textures. Each video texture was geerated by paig a 4 4 widow across the origial UCLA video. Four pas (two left ad two right) were geerated for each video sequece, resultig i a database of 8 paig textures, which we call the UCLA-pa database. The motio i this database is composed of both video textures ad camera paig, hece the dyamic texture is ot expected to perform well o it. Examples of the UCLA-pa database appear i Figure 3, ad video motages of both databases are available from [8]. 4

5 5.2. Experimetal setup A kerel dyamic texture was leared for each video i the database usig Algorithm ad a cetered Gaussia kerel with badwidth parameter σ 2, estimated for each video as σ 2 = 2 media{ y i y j 2 } i,j=,...,n () Both earest eighbor (NN) ad SVM classifiers [26] were traied usig the Marti distace for the kerel dyamic texture. The SVM used a RBF-kerel based o the Marti distace, k md (Θ a, Θ b ) = e 2σ 2 d2 (Θ,Θ 2). A oe-versusall scheme was used to lear the multi-class SVM problem, ad the C ad γ parameters were selected usig three-fold cross-validatio over the traiig set. We used the libsvm package [27] to trai ad test the SVM. For compariso, a NN classifier usig the Marti distace o the stadard dyamic texture [3] was traied, alog with a correspodig SVM classifier. NN ad SVM classifiers usig the image-space KL-divergece betwee dyamic textures [4] were also traied. Fially, experimetal results were averaged over four trials, where i each trial the databases were split differetly with 75% of data for traiig ad cross-validatio, ad 25% of the data for testig Results Figures 4 (a) ad (c) show the NN classifier performace versus, the umber of pricipal compoets (or the dimesio of the state space). While the NN classifier based o the kerel dyamic texture ad Marti distace (KDT-MD) performs similarly to the dyamic texture (DT-MD) o the UCLA database, KDT-MD outperforms DT-MD for all values of o the UCLA-pa database. The best icreases from 84.3% to 89.8.% o the UCLA-pa database whe usig KDT-MD istead of DT- MD, while oly icreasig from 89.% to 89.5% o the UCLA database. This idicates that the paig motio i the UCLA-pa database is ot well modeled by the dyamic texture, whereas the kerel dyamic texture has better success. The performace of the SVM classifiers is show i Figures 4 (b) ad (d). The dyamic texture ad kerel dyamic texture perform similarly o the UCLA database, with both improvig over their correspodig NN classifier. However, the KDT-MD SVM outperforms the DT-MD SVM o the UCLA-pa database (accuracies of 94.3% ad 92.8%, respectively). Whe lookig at the performace of the KL-based classifiers, we ote that for the UCLA databases the meaimage of the video is highly discrimiative for classificatio. This ca be see i Figures 4 (a), where the of the KL-divergece NN classifier is plotted for dyamic textures leared from the ormal data (DT-KL) ad from Database KDT-MD DT-MD DT-KL UCLA NN.895 (2).89 (5).365 (2) UCLA SVM.975 (2).965 (5).725 (2) UCLA-pa NN.898 (3).843 (3).86 (5) UCLA-pa SVM.943 (3).928 (25).92 (5) Table. Classificatio results for the UCLA ad UCLA-pa databases. () is the umber of pricipal compoets. zero-mea data (DT-KL). The best performace for DT- KL occurs whe =, i.e. the video is simply modeled as the mea image with some i.i.d. Gaussia oise. O the other had, whe the mea is igored i DT-KL, the performace of the classifier drops dramatically (from 94% to 5% for = ). Hece, much of the discrimiative power of the KL-based classifier comes from the similarity of image meas, ot from video motio. Because the Marti distace does ot use the image meas, we preset the classificatio results for DT-KL to facilitate a fair compariso betwee the classifiers. The DT-KL NN classifier performed worse tha both KDT-MD ad DT-MD, as see i Figures 4 (a) ad (c). The SVM traied o DT-KL improved the performace over the DT-KL NN classifier, but is still iferior to the KDT-MD SVM classifiers. A summary of the results o the UCLA ad UCLA-pa databases is give i Table. Fially, Figure 5 shows the distace matrix for DT- MD ad KDT-MD for three classes from UCLA-pa: two classes of water fallig from a ledge, ad oe class of boilig water (see Figure 5 (right) for examples). The DT-MD performs poorly o may of these sequeces because the water motio is chaotic, i.e. there are may discotiuities i the pixel values. O the other had, KDT-MD models the discotiuities, ad hece ca distiguish betwee the differet types of chaotic water motio. Ackowledgmets The authors thak Bejami Recht for helpful discussios, ad Giafraco Doretto ad Stefao Soatto for the database from [3]. This work was partially fuded by NSF award IIS ad NSF IGERT award DGE Refereces [] G. Doretto, A. Chiuso, Y. N. Wu, ad S. Soatto, Dyamic textures, Itl. J. Computer Visio, vol. 5, o. 2, pp. 9 9, 23. [2] A. B. Cha ad N. Vascocelos, Layered dyamic textures, i Neural Iformatio Processig Systems 8, 26, pp. 23. [3] P. Saisa, G. Doretto, Y. Wu, ad S. Soatto, Dyamic texture recogitio, i IEEE Cof. CVPR, vol. 2, 2, pp [4] A. B. Cha ad N. Vascocelos, Probabilistic kerels for the classificatio of auto-regressive visual processes, i CVPR, 25. [5] S. V. N. Vishwaatha, A. J. Smola, ad R. Vidal, Biet-cauchy kerels o dyamical systems ad its applicatio to the aalysis of dyamic scees, IJCV, vol. 73, o., pp. 95 9, 27. 5

6 UCLA NN UCLA SVM UCLA pa NN UCLA pa SVM KDT MD NN DT MD NN DT KL NN DT KL NN KDT MD SVM DT MD SVM DT KL SVM DT KL SVM KDT MD NN DT MD NN DT KL NN DT KL NN KDT MD SVM DT MD SVM DT KL SVM DT KL SVM (a) (b) (c) (d) Figure 4. Classificatio results o the UCLA database usig: (a) earest eighbors ad (b) SVM classifiers; ad results o the UCLA-pa database usig: (c) earest eighbors ad (d) SVM classifiers. Classifier is plotted versus the umber of pricipal compoets. boilig b ear wfalls a ear wfalls c ear Dyamic Texture boilig b ear wfalls a ear wfalls c ear boilig b ear wfalls a ear wfalls c ear Kerel Dyamic Texture boilig b ear wfalls a ear wfalls c ear Figure 5. Misclassificatio of chaotic water: the Marti distace matrices for three water classes usig (left) dyamic textures, ad (middle) kerel dyamic textures. Nearest eighbors i each row are idicated by a black dot, ad the misclassificatios are circled. (right) Four examples from each of the three water classes [6] G. Doretto, D. Cremers, P. Favaro, ad S. Soatto, Dyamic texture segmetatio, i IEEE ICCV, vol. 2, 23, pp [7] R. Vidal ad A. Ravichadra, Optical flow estimatio & segmetatio of multiple movig dyamic textures, i CVPR, 25. [8] A. B. Cha ad N. Vascocelos, Mixtures of dyamic textures, i IEEE Itl. Cof. Computer Visio, vol., 25, pp [9] L. Yua, F. We, C. Liu, ad H.-Y. Shum, Sythesizig dyamic textures with closed-loop liear dyamic systems, i Euro. Cof. Computer Visio, 24, pp [] C.-B. Liu, R.-S. Li, N. Ahuja, ad M.-H. Yag, Dyamic texture sythesis as oliear maifold learig ad traversig, i British Machie Visio Cof., vol. 2, 26, pp [] C.-B. Liu, R.-S. Li, ad N. Ahuja, Modelig dyamic textures usig subspace mixtures, i ICME, 25, pp [2] G. Doretto ad S. Soatto, Towards pleoptic dyamic textures, i 3rd Itl. Work. Texture Aalysis ad Sythesis, 23, pp [3] B. Schölkopf, A. Smola, ad K. R. Müller, Noliear compoet aalysis as a kerel eigevalue problem, Neural Computatio, vol., o. 5, pp , 998. [4] R. Duda, P. Hart, ad D. Stork, Patter Classificatio. Joh Wiley ad Sos, 2. [5] B. Schölkopf, S. Mika, A. Smola, G. Rätsch, ad K. R. Müller, Kerel pca patter recostructio via approximate pre-images, i ICANN, Perspectives i Neural Computig, 998, pp [6] S. Mika, B. Schölkopf, A. Smola, K. R. Müller, M. Scholz, ad G. Rätsch, Kerel PCA ad de-oisig i feature spaces, i Neural Iformatio Processig Systems, vol., 999, pp [7] J.-Y. Kwok ad I.-H. Tsag, The pre-image problem i kerel methods, IEEE Tras. Neural Networks, vol. 5, o. 6, 24. [8] A. B. Cha ad N. Vascocelos, Supplemetal material for classifyig video with kerel dyamic textures, Statistical Visual Computig Lab, Tech. Rep. SVCL-TR-27-3, 27, [9] Z. Ghahramai ad S. T. Roweis, Learig oliear dyamical systems usig a EM algorithm, i NIPS, 998. [2] H. Valpola ad J. Karhue, A usupervised esemble learig method for oliear dyamic state-space models, Neural Computatio, vol. 4, o., pp , 22. [2] J. Ham, D. D. Lee, S. Mika, ad B. Schölkopf, A kerel view of the dimesioality reductio of maifolds, i ICML, 24. [22] A. Rahimi, B. Recht, ad T. Darrell, Learig appearace maifolds from video, i IEEE CVPR, vol., 25, pp [23] A. Rahimi ad B. Recht, Estimatig observatio fuctios i dyamical systems usig usupervised regressio, i NIPS, 26. [24] R. J. Marti, A metric for ARMA processes, IEEE Trasactios o Sigal Processig, vol. 48, o. 4, pp. 64 7, April 2. [25] K. D. Cock ad B. D. Moor, Subspace agles betwee liear stochastic models, i IEEE Cof. Decisio ad Cotrol, 2. [26] V. N. Vapik, The ature of statistical learig theory. Spriger- Verlag, 995. [27] C.-C. Chag ad C.-J. Li, LIBSVM: a library for support vector machies, 2, cjli. 6