Learning a Locality Preserving Subspace for Visual Recognition

Transcription

1 Learnng a Localty Preservng Subspace for Vsual Recognton Xaofe He *, Shucheng Yan #, Yuxao Hu, and Hong-Jang Zhang Mcrosoft Research Asa, Bejng , Chna * Department of Computer Scence, Unversty of Chcago (xaofe@cs.uchcago.edu) # School of Mathematcal Scence, Pekng Unversty, Chna Abstract Prevous orks have demonstrated that the face recognton performance can be mproved sgnfcantly n lo dmensonal lnear subspaces. Conventonally, prncpal component analyss (PCA) and lnear dscrmnant analyss (LDA) are consdered effectve n dervng such a face subspace. Hoever, both of them effectvely see only the Eucldean structure of face space. In ths paper, e propose a ne approach to mappng face mages nto a subspace obtaned by Localty Preservng Projectons (LPP) for face analyss. We call ths Laplacanface approach. Dfferent from PCA and LDA, LPP fnds an embeddng that preserves local nformaton, and obtans a face space that best detects the essental manfold structure. In ths ay, the unanted varatons resultng from changes n lghtng, facal expresson, and pose may be elmnated or reduced. We compare the proposed Laplacanface approach th egenface and fsherface methods on three test datasets. Expermental results sho that the proposed Laplacanface approach provdes a better representaton and acheves loer error rates n face recognton. 1. Introducton In recent years, computer vson research has tnessed a grong nterest n subspace analyss technques [1][6][14][16][0][1]. A face mage can be represented as a pont n the mage space (gven by the number of pxels n the mage). Before e utlze any classfcaton technque, t s benefcal to frst perform dmensonalty reducton to project an mage nto a lo dmensonal feature space or so-called face space, due to the consderaton of learnablty and computatonal effcency. Specfcally, learnng from examples s computatonally nfeasble f t has to rely on hgh-dmensonal representatons. In partcular, Prncpal Component Analyss (PCA) [16] and Lnear Dscrmnant Analyss (LDA) [1] have been appled to face recognton th mpressve results. PCA s an egenvector method desgned to model lnear varaton n hgh-dmensonal data. PCA performs dmensonalty reducton by projectng the orgnal n- dmensonal data onto the k(<<n)-dmensonal lnear subspace spanned by the leadng egenvectors of the data s covarance matrx. Its goal s to fnd a set of mutually orthogonal bass functons that capture the drectons of maxmum varance n the data and for hch the coeffcents are parse decorrelated. For lnearly embedded manfolds, PCA s guaranteed to dscover the dmensonalty of the manfold and produces a compact representaton. LDA s a supervsed learnng algorthm. LDA searches for the projecton axes on hch the data ponts of dfferent classes are far from each other and at the same tme here the data ponts of a same class are close to each other. Unlke PCA hch encodes nformaton n an orthogonal lnear space, LDA encodes dscrmnatng nformaton n a lnear separable space hose bases are not necessarly orthogonal. Recently, a number of research efforts have shon that the face mages possbly resde on a nonlnear submanfold [9][10][11][15]. Hoever, both PCA and LDA effectvely see only the Eucldean structure. hey fal to dscover the underlyng structure, f the face mages le on a nonlnear submanfold hdden n the mage space. Some nonlnear technques have been proposed to dscover the nonlnear structure of the manfold,.e. Isomap [15], LLE [9], and Laplacan egenmaps []. hese nonlnear methods do yeld mpressve results on some benchmark artfcal data sets. Hoever, they yeld maps that are defned only on the tranng data ponts and ho to evaluate the maps on ne testng ponts remans unclear. herefore, these nonlnear manfold learnng technques mght not be sutable for some computer vson tasks, such as face recognton. In the meantme, there has been some nterest n the problem of developng lo dmensonal representatons through kernel based technques for face recognton [5][19]. hese methods can dscover the nonlnear structure of the face mages. Hoever, they are computatonally expensve. Moreover, none of them explctly consders the structure of the manfold on hch the face mages possbly resde. In ths paper, e propose a ne approach to face representaton and recognton, hch explctly consders the face manfold structure. o be specfc, an adjacency graph s constructed to model the local structure of the face manfold. A Localty Preservng Subspace for face representaton s learned by usng Localty Preservng Projectons (LPP). Each face mage n the mage space s mapped to a lo-dmensonal face subspace, hch s Proceedngs of the Nnth IEEE Internatonal Conference on Computer Vson (ICCV 003) -Volume Set

2 characterzed by a set of feature mages, called Laplacanfaces. he face subspace preserves local structure, and thus has more dscrmnatng poer than egenfaces from the classfcaton vepont. Moreover, the localty preservng property makes our algorthm nsenstve to the unanted varatons due to changes n lghtng, facal expresson, and veng ponts. It s orthhle to hghlght several aspects of the proposed approach here: 1. Whle PCA ams to preserve the global structure of the mage space, and LDA ams to preserve the dscrmnatng nformaton; LPP ams to preserve the local structure of the mage space. In many real orld classfcaton problems, the local manfold structure s more mportant than the global Eucldean structure, especally hen nearest-neghbor lke classfers are used for classfcaton.. An effcent subspace learnng algorthm for face recognton should be able to detect the nonlnear manfold structure of the face space. Our proposed Laplacanface method explctly consders the manfold structure hch s modeled by an adjacency graph. 3. LPP shares some smlar propertes th LLE, such as localty preservng character. Hoever, ther objectve functons are totally dfferent. LPP s obtaned by fndng the optmal lnear approxmatons to the egenfunctons of the Laplace Beltram operator on the manfold [][4]. LPP s lnear, hle LLE s nonlnear. Moreover, LPP s defned everyhere, hle LLE s defned only on the tranng data ponts and t s unclear ho to evaluate the map for ne test ponts. In contrast, LPP may be smply appled to any ne data pont to locate t n the reduced representaton space. he rest of ths paper s organzed as follos: Secton descrbes the objectve functons of PCA and LDA. he Localty Preservng Projecton algorthm s descrbed n secton 3. In secton 4, e present the manfold ays of face analyss. he expermental results are shon n Secton 5. Fnally, e gve concludng remarks and future ork n Secton 6.. PCA and LDA One approach to copng th the problem of excessve dmensonalty of the mage space s to reduce the dmensonalty by combnng features. Lnear combnatons are partcular attractve because they are smple to compute and analytcally tractable. In effect, lnear methods project the hgh-dmensonal data onto a loer dmensonal subspace. Consderng the problem of representng all of the vectors n a set of n d-dmensonal samples x 1, x,, x n, th zero mean, by a sngle vector y={y 1, y,, y n } such that y represent x. Specfcally, e fnd a lnear mappng from the d-dmensonal space to a lne. Wthout loss of generalty, e denote the transformaton vector by. hat s, x = y. Actually, the magntude of s of no real sgnfcance, because t merely scales y. In face recognton, each vector x denotes a face mage. Dfferent objectve functons ll yeld dfferent algorthms th dfferent propertes. PCA seeks a projecton that best represents the data n a least-squares sense. he matrx s a projecton onto the prncpal component space spanned by {} hch mnmzes the follong objectve functon, mn n = 1 x x he output set of prncpal vectors 1,,, k are an orthonormal set of vectors representng the egenvectors of the sample covarance matrx assocated th the k < d largest egenvalues. Whle PCA seeks drectons that are effcent for representaton, LDA seeks drectons that are effcent for dscrmnaton. Suppose e have a set of n d-dmensonal samples x 1, x,, x n, belongng to l classes of faces. he objectve functon s as follos, S B max S S S B W = = l = 1 l = 1 C C W ( ) ( ) ( m m)( m m) E ( ) ( ) ( ) ( ) [( x m )( x m ) ] here m s the total sample mean vector, C s the number of samples n class C, m () are the average vectors of C, and x () are the sample vectors assocated to C. We call S W the thn-class scatter matrx and S B the beteenclass scatter matrx. 3. Learnng a Localty Preservng Subspace Both PCA and LDA am to preserve the global structure. Hoever, n many real orld applcatons, the local structure s more mportant, especally hen nearestneghbor search needs to be performed. In ths secton, e descrbe ho to learn a Localty Preservng Subspace by usng Localty Preservng Projectons (LPP) [4]. LPP s a lnear approxmaton of the nonlnear Laplacan Egenmaps []. It seeks to preserve the ntrnsc geometry of the data and the local structures. he objectve functon of LPP s as follos: mn ( y y j ) Sj y j he objectve functon th our choce of symmetrc eghts S j (S j = S j ) ncurs a heavy penalty f neghborng ponts x and x j are mapped far apart. herefore, mnmzng t s an attempt to ensure that f x and x j are close then y and y j are close as ell. S j can be thought of as a smlarty measure beteen objects. Let denote the Proceedngs of the Nnth IEEE Internatonal Conference on Computer Vson (ICCV 003) -Volume Set

3 transformaton vector. By smple algebra formulaton, e can reduce the above objectve functon as follos: = = 1 j = = x D j ( y y ) ( x x ) S XLX j x X ( D S) X S j j j j x S here X = [x 1, x,, x n ], and D s a dagonal matrx; ts entres are column (or ro, snce S s symmetrc) sums of S, D = j S j.. L = D S s the Laplacan matrx [3]. Matrx D provdes a natural measure on the data ponts. he bgger the value D (correspondng to y ) s, the more mportant s y. herefore, e mpose a constrant as follos: y Dy = 1 XDX = 1 Fnally, the mnmzaton problem reduces to fndng: arg mn XDX =1 XLX he transformaton vector that mnmzes the objectve functon s gven by the mnmum egenvalue soluton to the generalzed egenvalue problem: XLX = λxdx Note that the to matrces XLX and XDX are both symmetrc and postve sem-defnte. he dervaton reflects the ntrnsc geometrc structure of the manfold. he theoretcal justfcaton for LPP can be traced back to [4]. 4. Manfold Ways of Face Analyss In the above to sectons, e have descrbed three dfferent lnear subspace learnng algorthm. he key dfference beteen PCA, LDA and LPP s that, PCA and LDA am to dscover Eucldean structure, hle LPP ams to dscover manfold structure. In ths Secton, e dscuss the manfold ays of face analyss Manfold Learnng va Dmensonalty Reducton In many cases, face mages may be vsualzed as ponts dran on a lo-dmensonal manfold hdden n a hghdmensonal Eucldean space. Specally, e can consder that a sheet of rubber s crumpled nto a (hgh dmensonal) j x j ball. he objectve of a dmensonalty-reducng mappng s to unfold the sheet and to make ts lo-dmensonal structure explct. If the sheet s not torn n the process, the mappng s topology-preservng. Moreover, f the rubber s not stretched or compressed, the mappng preserves the metrc structure of the orgnal space. In ths paper, our objectve s to dscover the face manfold by a localty-preservng mappng for face representaton and recognton. 4.. Learnng Laplacanfaces for Representaton In secton 3, e have descrbed LPP, a method for learnng a localty preservng subspace. It s obtaned by fndng the optmal lnear approxmatons to the egenfunctons of the Laplace Betram operator on the manfold [4]. Base on LPP, e descrbe our Laplacanface method for face representaton and recognton. In the face analyss and recognton problems one s confronted th the dffculty that the matrx XDX s sometmes sngular. hs stems from the fact that, sometmes the number of mages n the tranng set (m) s much smaller than the number of pxels n each mage (n). In such case, the rank of XDX s at most m, hle XDX s an n n matrx, hch mples that XDX s sngular. o overcome the complcaton of a sngular XDX, e frst project the mage set to a PCA subspace so that the resultng matrx XDX s nonsngular. Another consderaton of usng PCA as preprocessng s for nose reducton. hs method, e call Laplacanface, can learn an optmal subspace for face representaton and recognton. he algorthmc procedure of Laplacanface s formally stated belo: 1. PCA projecton: We project the mage set {x } nto the PCA subspace by throng aay the smallest prncpal components. In our experments, e kept 98% nformaton n the sense of reconstructon error. For the sake of smplcty, e stll use x to denote the mages n the PCA subspace n the follong steps. We denote the transformaton matrx of PCA by W PCA.. Constructng the nearest-neghbor graph: Let G denote a graph th n nodes. he th node corresponds to the face mage x. We put an edge beteen nodes and j f x and x j are close,.e. x s among k nearest neghbors of x or x s among k nearest neghbors of x j. Note that, one mght take a more utltaran perspectve and construct a nearest-neghbor graph based on the class labels. hat s, e put an edge beteen to nodes f and only f they have the same class label. he constructed nearest-neghbor graph s an approxmaton of the local manfold structure. 3. Choosng the eghts: If node and j are connected, put Proceedngs of the Nnth IEEE Internatonal Conference on Computer Vson (ICCV 003) -Volume Set

4 Fgure 1. o-dmensonal lnear embeddng of face mages by Localty Preservng Projecton. As can be seen, the face mages are dvded nto to parts, the faces th open mouth and the faces th closed mouth. Moreover, t can be clearly seen that the pose and expresson of human faces change contnuously and smoothly, from top to bottom, from left to rght. he bottom mages correspond to ponts along the rght path (lnked by sold lne), llustratng one partcular mode of varablty n pose. S j = e x x t here t s a sutable constant. Otherse, put S j = 0. he eght matrx S of graph G models the face manfold structure by preservng local structure. he justfcaton for ths choce of eghts can be traced back to []. 4. Egenmap: Compute the egenvectors and egenvalues for the generalzed egenvector problem: j XLX = λxdx (1) here D s a dagonal matrx hose entres are column (or ro, snce S s symmetrc) sums of S, D = S j j. L = D S s the Laplacan matrx. he th column of the matrx X s x. Let 0, 1,, k-1 be the solutons of equaton (1), ordered accordng to ther egenvalues, λ 0 <λ 1 < <λ k-1. hus, the embeddng s as follos: x y = W x W = W PCA W LPP WLPP [ 0, 1, L, 1 ] = k here y s a k-dmensonal vector. W s the transformaton matrx. hs lnear mappng best preserves the manfold s estmated ntrnsc geometry n a lnear sense. he column vectors of W are the so called Laplacanfaces hch span the face subspace. 4.3 Face Manfold Analyss No consder a smple example of mage varablty, a set of face mages are generated hle the human face rotates sloly. Intutvely, the set of face mages correspond to a contnuous curve n mage space, snce there s only one degree of freedom,.e. the angel of rotaton. hus, e can say that the set of face mages are ntrnscally one-dmensonal. Actually, much recent ork [9][10][11][15] has shon that the face mages do resde on a lo-dmensonal submanfold embedded n hghdmensonal mage space. herefore, an effectve subspace learnng algorthm should be able to detect the nonlnear manfold structure. he conventonal algorthms, such as PCA and LDA, model the face mages n Eucldean space. hey effectvely see only the Eucldean struc- Proceedngs of the Nnth IEEE Internatonal Conference on Computer Vson (ICCV 003) -Volume Set

5 Fgure 3. he frst 10 Egenfaces (frst ro), Fsherfaces (second ro) and Laplacanfaces (thrd ro) calculated from the face mages n the YALE database. Fgure. he left plots sho the results of PCA. he rght plots sho the results of LPP. he frst bass s shon as a longer lne segment, and the second bass s shon as a shorter lne segment. Clearly, LPP has more dscrmnatng poer than PCA, and s less senstve to outlers. ture; thus, they fal to detect the ntrnsc lo dmensonalty. Wth neghborhood preservng character, the LPP algorthm s capable of capturng the ntrnsc manfold structure to a large extent. Fgure 1 shos an example that the face mages th varous pose and expresson of a person are mapped nto a to-dmensonal subspace. he face mage data set used here s the same as that used n [9]. he representatve face mages are shon n the dfferent parts of the space. As can be seen, the face mages are dvded nto to parts. he left part ncludes the face mages th open mouth, and the rght part ncludes the face mages th closed mouth. hs s because that, by tryng to preserve local structure n the embeddng, LPP mplctly emphaszes the natural clusters n the data. Specfcally, t makes the neghborng ponts n the mage space nearer n the face space, and faraay ponts n the mage space farther n the face space. Some theoretcal analyss can be found n [][4][1]. Moreover, e can see from the fgure that the pose and expresson of the faces change contnuously and smoothly. he bottom mages correspond to ponts along the rght path (lnked by sold lne), llustratng one partcular mode of varablty n pose. hs observaton tells us that LPP s capable of capturng the ntrnsc face manfold structure. 5. Expermental Results In ths secton, several experments are carred out to sho the effectveness of our proposed Laplacanface method for face representaton and recognton. We begn th to smple synthetc examples to compare LPP and PCA. 5.1 Smple Synthetc Examples o smple synthetc examples are gven n Fg.. Both of the to data sets correspond to an essentally onedmensonal manfold. Projecton of the data ponts onto the frst bass ould then correspond to a one-dmensonal lnear manfold representaton. he second bass, shon as a shorter lne segment n the fgure, ould be dscarded n ths lo-dmensonal example. As can be seen, PCA captures the drecton of maxmum varance n the data. LPP fnds drecton hch preserves local structure and the dscrmnatng poer. Moreover, PCA s senstve to outlers hle LPP s not. 5. Face Representaton Usng Laplacanfaces As e descrbed prevously, a face mage can be represented as a pont n mage space. A typcal mage of sze m n descrbes a pont n m n-dmensonal mage space. Hoever, due to the unanted varatons resultng from changes n lghtng, facal expresson, and pose, the mage space mght not be an optmal space for vsual representaton and recognton. In secton, e have dscussed ho to learn a localty preservng face subspace hch s nsenstve to outler and nose. he mages of faces n the tranng set are used to learn such a face subspace. he subspace s spanned by the Laplacanfaces as descrbed n secton 4.. We can dsplay the Laplacanfaces as a sort of feature mages. Usng the Yale face database as the tranng set, e present the frst 10 Laplacanfaces n Fgure 3, together th Proceedngs of the Nnth IEEE Internatonal Conference on Computer Vson (ICCV 003) -Volume Set

6 Fgure 4. he sample cropped face mages of one ndvdual from PIE database. he orgnal face mages are taken under varyng pose, llumnaton, and expresson. egenfaces and fsherfaces. hus, a face mage can be mapped nto the localty preservng subspace spanned by the Laplacanfaces. 5.3 Face Recognton Once the Laplacanfaces are created, face recognton [1][16][17] becomes a pattern classfcaton task. In ths secton, e nvestgate the performance of our proposed Laplacanface method for face recognton. he system performance s compared th the egenface method [16] and the fsherface method [1], to of the most popular methods n face recognton. In ths study, three face databases ere tested. he frst one s the Yale database [18], the second one s the PIE (pose, llumnaton, and expresson) database from CMU [13], and the thrd one s the MSRA database collected by our on. In all the experments, preprocessng to locate the faces as appled. Orgnal mages ere normalzed (n scale and orentaton) such that the to eyes ere algned at the same poston. hen, the facal areas ere cropped nto the fnal mages for matchng. he sze of each cropped mage n all the experments s 3 3 pxels, th 56 grey levels per pxel. hus, each mage can be represented by a 104-dmensonal vector n mage space. No further preprocessng s done. Fgure 5 shos an example of the orgnal face mage and the cropped mage. Dfferent pattern classfers have been appled for face recognton, ncludng nearest-neghbor [16], Bayesan [7], and support vector machne [8], etc. In ths paper, e apply nearest-neghbor classfer for ts smplcty. he recognton process has three steps. Frst, e calculate the Laplacanfaces from the tranng set of face mages; then, the ne face mage to be dentfed s projected nto the face subspace spanned by the Laplacanfaces; fnally, the ne face mage s dentfed by a nearest-neghbor classfer. For Yale and PIE database, a random subset of a fxed sze s taken th labels to form the tranng set. he rest of the database s consdered to be the testng set Yale Database he Yale face database [18] s constructed at the Yale Center for Computatonal Vson and Control. It contans 165 grayscale mages of 15 ndvduals. he mages demonstrate varatons n lghtng condton (left-lght, centerlght, rght-lght), facal expresson (normal, happy, sad, Fgure 5. he orgnal face mage and the cropped mage. sleepy, surprsed, and nk), and th/thout glasses. For each ndvdual, 6 faces are used for tranng, and the rest 5 are used for testng. he face subspace s constructed by our Laplacanfaces method to best preserve the local structure hle reducng the dmensonalty of the mage space. For each face mage, t can be projected nto the face subspace by the transformaton matrx W,.e. Laplacanfaces. he recognton results are shon n able 1. It s found that the Laplacanface approach sgnfcantly outperforms both egenface and fsherface approaches. he error rate s 11.3%, 0.0% and 5.3% for Laplacanface, fsherface, and egenface methods, respectvely. he correspondng face subspaces are called optmal face subspaces for each method. here s no sgnfcant mprovement f more dmensons are used. Fgure 7 shos a plot of error rate vs. dmensonalty reducton. Note that, the upper bound of the dmensonalty of fsherfaces s c-1 here c s the number of ndvduals PIE Database he CMU PIE face database contans 68 subjects th 41,368 face mages as a hole. he face mages ere captured by 13 synchronzed cameras and 1 flashes, under varyng pose, llumnaton and expresson. We use 170 near frontal face mages for each ndvdual n our experment, 85 for tranng and the other 85 for testng. Fgure 4 shos some of the faces th pose, llumnaton and expresson varatons n the PIE database. able shos the recognton results. As can be seen, Laplacanface method performed better than egenface and fsherface methods. Fgure 8 shos a plot of error rate vs. dmensonalty reducton MSRA Database hs database as collected at the Mcrosoft Research Asa. It contans 1 ndvduals, captured n to dfferent sessons th dfferent backgrounds and llumnatons. 64 to 80 face mages are collected for each ndvdual n each sesson. All the faces are frontal. Fgure 6 shos the sample cropped face mages from ths database. In ths test, one sesson s used for tranng and the other s used for testng. able 3 shos the recognton results. Laplacanface approach has loer error rate (8.%) than those of egenface (35.4%) and fsherface (6.5%). Fgure 9 shos a plot of error rate vs. dmensonalty reducton. Proceedngs of the Nnth IEEE Internatonal Conference on Computer Vson (ICCV 003) -Volume Set

7 Fgure 6. he sample cropped face mages of 8 ndvduals from MSRA database. he face mages n the frst ro are taken n the frst sesson, hch are used for tranng. he face mages n the second ro are taken n the second sesson, hch are used for testng. he to mages n the same column are correspondng to the same ndvdual Dscussons hree experments have been systematcally performed. hese experments reveal a number of nterestng ponts: 1. All these three approaches performed better n the optmal face subspace than n the orgnal mage space.. In all the three experments, the Laplacanface approach consstently performed better than the egenface and fsherface approaches. Especally, t sgnfcantly outperformed the fsherface and egenface approaches on Yale database and MSRA database. 3. hough Laplacanface does not explctly consder the classfcaton problem, t stll outperforms fsherfaces, hch s based on dscrmnant analyss. hs s because that, Laplacanface approach encodes more dscrmnatng nformaton n the lodmensonal face subspace by preservng local structure hch s more mportant than the global structure for classfcaton, especally hen nearest neghbor lke classfers are used. In fact, f there s a reason to beleve that Eucldean dstances ( x x j ) are meanngful only f they are small (local), then the LPP algorthm fnds a projecton that respects such a belef. Another reason s that, as e sho n Fg. 1, the face mages probably resde on a nonlnear manfold. herefore, an effcent and effectve subspace representaton of face mages should be capable of characterng the nonlnear manfold structure, hle the Laplacanfaces are exactly derved by fndng the optmal lnear approxmatons to the egenfunctons of the Laplace Beltram operator on the face manfold [][4]. By dscoverng the face manfold structure, our Laplacanface approach can dentfy the person th varous pose, llumnaton and expresson. 4. he Laplacanface approach appears to be the best at smultaneously handlng varaton n lghtng, pose and expresson. 6. Concluson and Future Work he manfold ays of face representaton and recognton s ntroduced n ths paper n order to detect the underlyng nonlnear manfold structure n the manner of subspace learnng. o the best of our knoledge, ths s the frst devoted ork on face representaton and recognton hch explctly consders manfold structure n a lnear manner. he manfold structure s approxmated by the nearest-neghbor graph computed from the data ponts. Usng the noton of the Laplacan of the graph, e then compute a transformaton matrx hch maps the face mages nto the face subspace. We call ths Laplacanfaces approach. he Laplacanfaces are obtaned by fndng the optmal lnear approxmatons to the egenfunctons of the Laplace Beltram operator of the face manfold [][4]. hs lnear transformaton optmally preserves local manfold structure. Expermental results on the Yale database, CMU PIE database, and MSRA database sho the effectveness of our method. One of the central problems n face manfold learnng s to estmate the ntrnsc dmensonalty of the nonlnear manfold, or, degrees of freedom. Moreover, by usng kernel methods, the lnear projectve maps can be easly extended to nonlnear maps,.e. kernel Laplacanfaces hch mght be able to detect the nonlnear face manfold structure. We are currently explorng these problems n theory and practce. able 1. Performance comparson on the Yale database Approach Dms Error Rate Egenfaces % Fsherfaces % Laplacanfaces % able. Performance comparson on the PIE database Approach Dms Error Rate Egenfaces % Fsherfaces % Laplacanfaces % able 3. Performance comparson on MSRA database Approach Dms Error Rate Egenfaces % Fsherfaces % Laplacanfaces 66 8.% Proceedngs of the Nnth IEEE Internatonal Conference on Computer Vson (ICCV 003) -Volume Set

8 Fgure 7. Error rate vs. dmensonalty reducton on Yale database Fgure 8. Error rate vs. dmensonalty reducton on Pe database Fgure 9. Error rate vs. dmensonalty reducton on Our On database References [1] P. N. Belhumeur, J. P. Hespanha and D. J. Kregman, Egenfaces vs. Fsherfaces: Recognton Usng Class Specfc Lnear Projecton, IEEE rans. Pattern Analyss and Machne Intellgence, vol. 19, No. 7, 1997, pp [] M. Belkn and P. Nyog, Laplacan Egenmaps and Spectral echnques for Embeddng and Clusterng, Advances n Neural Informaton Processng System 15, Vancouver, Brtsh Columba, Canada, 001. [3] Fan R. K. Chung, Spectral Graph heory, Regonal Conferences Seres n Mathematcs, number 9, [4] Xaofe He and Partha Nyog, Localty Preservng Projecton, echncal Report R-00-09, Department of Computer Scence, the Unversty of Chcago. [5] Q. Lu, R. Huang, H. Lu, and S. Ma, Face Recognton Usng Kernel Based Fsher Dscrmnant Analyss, n Proc. of the ffth Int. Conf. on Automatc Face and Gesture Recognton, Washngton D. C., May 00. [6] A. M. Martnez and A. C. Kak, PCA versus LDA, IEEE rans. on Pattern Analyss and Machne Intellgence, 3():8-33, 001. [7] B. Moghaddam and A. Pentland, Probablstc Vsual Learnng for Object Representaton, IEEE rans. on Pattern Analyss and Machne Intellgence, vol 19, pp , [8] P. J. Phllps, Support Vector Machnes Appled to Face Recognton. In Advances n Neural Informaton Processng Systems 11, pp , [9] Sam. Roes, and Larence K. Saul, Nonlnear Dmensonalty Reducton by Locally Lnear Embeddng, Scence, vol 90, December 000. [10] Sam Roes, Larence Saul and Geoff Hnton, Global Coordnaton of Local Lnear Models, Advances n Neural Informaton Processng System 14, 001. [11] H. Sebastan Seung and Danel D. Lee, he Manfold Ways of Percepton, Scence, vol 90, December 000. [1] J. Sh and J. Malk, Normalzed Cuts and Image Segmentaton, IEEE rans. on Pattern Analyss and Machne Intellgence, (000), [13]. Sm, S. Baker, and M. Bsat, he CMU Pose, Illumnaton, and Expresson (PIE) Database, n Proceedngs of the IEEE Internatonal Conference on Automatc Face and Gesture Recognton, May, 00. [14] L. Srovch and M. Krby, Lo-Dmensonal Procedure for the Characterzaton of Human Faces, Journal of the Optcal Socety of Amerca A, vol. 4, pp , [15] Joshua B. enenbaum, Vn de Slva, and Johh C. Langford, A Global Geometrc Frameork for Nonlnear Dmensonalty Reducton,Scence, vol 90, December 000. [16] M. urk and Pentland, Face Recognton Usng Egenfaces, n Proc. IEEE Internatonal Conference on Computer Vson and Pattern Recognton, Mau, Haa, [17] L. Wskott, J.M. Fellous, N. Kruger, and C.v.d. Malsburg, Face Recognton by Elastc Bunch Graph Matchng, IEEE rans. on Pattern Analyss and Machne Intellgence, 19: , [18] Yale Unv. Face Database, [19] Mng-Hsuan Yang, Kernel Egenfaces vs. Kernel Fsherfaces: Face Recognton Usng Kernel Methods, n Proc. of the ffth Int. Conf. on Automatc Face and Gesture Recognton, Washngton D. C., May 00. [0] J. Yang, Y. Yu, and W. Kunz, An Effcent LDA Algorthm for Face Recognton, he sxth Internatonal Conference on Control, Automaton, Robotcs and Vson, Sngapore, 000. [1] W. Zhao, R. Chellappa, and P. J. Phllps, Subspace Lnear Dscrmnant Analyss for Face Recognton, echncal Report CAR-R-914, Center for Automaton Research, Unversty of Maryland, Proceedngs of the Nnth IEEE Internatonal Conference on Computer Vson (ICCV 003) -Volume Set