LEARNING A WARPED SUBSPACE MODEL OF FACES WITH IMAGES OF UNKNOWN POSE AND ILLUMINATION

Transcription

1 LEARNING A WARPED SUBSPACE MODEL OF FACES WITH IMAGES OF UNKNOWN POSE AND ILLUMINATION Jhun Hamm, and Danel D. Lee GRASP Laboratory, Unversty of Pennsylvana, 3330 Walnut Street, Phladelpha, PA, USA jhham@seas.upenn.edu, ddlee@seas.upenn.edu Keywords: Abstract: mage-based modelng, probablstc generatve model, llumnaton subspace, super-resoluton, MAP estmaton, multscale mage regstraton In ths paper we tackle the problem of learnng the appearances of a person s face from mages wth both unknown pose and llumnaton. The unknown, smultaneous change n pose and llumnaton makes t dffcult to learn 3D face models from data wthout manual labelng and trackng of features. In comparson, mage-based models do not requre geometrc knowledge of faces but only the statstcs of data tself, and therefore are easer to tran wth mages wth such varatons. We take an mage-based approach to the problem and propose a generatve model of a warped llumnaton subspace. Image varatons due to llumnaton change are accounted for by a low-dmensonal lnear subspace, whereas varatons due to pose change are approxmated by a geometrc warpng of mages n the subspace. We demonstrate that ths model can be effcently learned va MAP estmaton and multscale regstraton technques. Wth ths learned warped subspace we can jontly estmate the pose and the lghtng condtons of test mages and mprove recognton of faces under novel poses and llumnatons. We test our algorthm wth synthetc faces and real mages from the CMU PIE and Yale face databases. The results show mprovements n predcton and recognton performance compared to other standard methods. 1 INTRODUCTION We tackle the problem of learnng generatve models of a person s face from mages wth both unknown pose and llumnaton. The appearance of a person s face undergoes large varatons as llumnaton and vewng drectons change. A full 3D model of a face allows us to synthesze mages at arbtrary poses and llumnaton condtons. However, learnng such a 3D model from mages alone s very dffcult snce t requres that feature correspondences are known accurately, even under dramatc lghtng changes. In ths paper we develop an mage-based approach, whch does not use 3D models nor solve correspondence problems, but nstead drectly learns the statstcal propertes of mages under pose and llumnaton varatons. (see Fg. 1). One of the smplest mage-based models of Fgure 1: Typcal unlabeled mages wth both varyng pose and llumnaton condtons make t dffcult to learn 3D structures drectly from sample mages. We am to learn an appearance model of a person s face gven such mages wthout fndng, labelng and trackng features between frames. faces s an Egenface model (Turk and Pentland, 1991), whch models data as an affne subspace n the space of pxel ntenstes. Although Egenfaces were orgnally appled to mage varatons across dfferent people, a subspace model can explan the llumnaton varaton of a sngle per-

2 n the subspace. A schematc of the warped subspace s depcted n Fg Related Work Fgure 2: A probablstc model of pose and llumnaton varatons. The ellpsod n the mddle represents frontal mages wth all possble llumnatons, lyng closely on a low-dmensonal subspace. As the vewpont changes from a rght-profle to a left-profle pose, the ellpsod s transported contnuously along a nonlnear manfold. We model ths nonlnear varaton wth a geometrc warpng of mages. son exceptonally well (Hallnan, 1994; Epsten et al., 1995). The so-called llumnaton subspace has been thoroughly studed theoretcally (Ramamoorth, 2002; Basr and Jacobs, 2003). However, such a lnear subspace model cannot cope wth the smultaneous nonlnear change of poses. On the other hand, suppose we are gven multple-pose vews of a face under a fxed lghtng condton. If the pose change s moderate, we can learn a generatve model by geometrcally regsterng the mult-vew mages and probablstcally combnng them to estmate the unknown latent mage. Such generatve models have been proposed for the super-resoluton problem (Harde et al., 1997; Tppng and Bshop, 2002; Capel and Zsserman, 2003). However, prevous work consders only a sngle latent mage rather than a latent subspace, and therefore can handle only one-dmensonal llumnaton changes and not the full range of llumnaton varatons from arbtrary lght sources. In ths paper we model the smultaneous change of pose and llumnaton of a person s face by a novel warped subspace model. Image varatons due to llumnaton change at a fxed pose are captured by a low-dmensonal llumnaton subspace; and varatons due to pose change are approxmated by a geometrc warpng of mages Image-based models of faces have been proposed before. A popular mult-pose representaton of mages s the lght-feld presentaton, whch models the radance of lght as a functon of the 5D pose of the observer (Gross et al., 2002a; Gross et al., 2002b; Zhou and Chellappa, 2004). Theoretcally, the lght-feld model provdes posenvarant recognton of mages taken wth arbtrary camera and pose when the llumnaton condton s fxed. Zhou et al. extended the lght-feld model to a blnear model whch allows smultaneous change of pose and llumnaton (Zhou and Chellappa, 2004). However, n practce, the camera pose of the test mage has to be known beforehand to compare t wth the pre-computed lghtfeld, whch effectvely amounts to knowng the correspondence n 3D model-based approaches. Ths model s also unable to extend to the representaton to a novel pose. In the super-resoluton feld, the dea of usng latent subspaces n generatve models has been suggested by (Capel and Zsserman, 2001; Gunturk et al., 2003). However the learned subspaces reflect mxed contrbutons from pose, llumnaton, subject denttes, etc. In our case the subspace encodes 3D structure, albedo and the lowpass flterng nature of the Lambertan reflectance functon (Basr and Jacobs, 2003), and the pose change s dedcated to geometrc transforms. Furthermore, we show how to learn the bass, pose and llumnaton condtons drectly and smultaneously from a few mages of both unknown pose and llumnaton. In our method we estmate the geometrc warpng varable va a contnuous optmzaton nstead of searchng over a lmted or fnte set of predefned transformatons (Harde et al., 1997; Frey and Jojc, 1999; Tppng and Bshop, 2002). The remander of the paper s organzed as follows. In Sec. 2, we formulate the warped subspace model n a probablstc generatve framework, and descrbe how to jontly estmate the pose and the llumnaton wth a known bass. In Sec. 3, we descrbe a maxmum a posteror (MAP) approach to the learnng of a bass as well as the estmaton of pose and llumnaton smultaneously, and explan how a pror dstrbuton and effcent optmzaton can be employed to learn the model. In Sec. 4, we perform recog-

3 nton experments on real data sets. We conclude wth dscussons n Sec JOINT ESTIMATION OF POSE AND ILLUMINATION In ths secton we explan the elements of generatve models of mages and optmzaton technques to jontly estmate pose and llumnaton. 2.1 Generatve Model of Mult-vew Images A smple generatve model common to superresoluton methods s the followng: x = W g z + ɛ, (1) where x s an observed mage, z s a latent mage, and W g s a warpng operator on the latent mage: let z(u, v) be the pxel ntensty at (u, v) R 2 and T g : R 2 R 2 be a transform of coordnates paramaterzed by g, then (W g z)(u, v) = z(tg 1 (u, v)). In the fnte-pxel doman, we assume x and z are D-dmensonal vectors of mage ntenstes where D s the number of pxels, and W g s a D D matrx representng the warpng and the subsequent nterpolaton. From the defnton above, the W g s a nonlnear functon of warpng varables g; however, t s a lnear functon of mages z, whch can be seen by the equalty W g (a 1 I 1 + a 2 I 2 ) = a 1 (W g I 1 ) + a 2 (W g I 2 ), for any two mages I 1 and I 2 and real numbers a 1 and a 2. We consder the perspectve transforms n ths paper, so g s a 8-dmensonal vector. The X = {x 1,..., x N } and G = {g 1,..., g N } denote aggregates of N observed mages and N warpng varables. If the nose ɛ s an ndependent, addtve Gaussan nose, that s p(x z, g) N (W g z, Ψ), then the log-lkelhood of the observed samples X becomes L = log p(x z, G) = log p(x z, g ) = 1 (x W g z) Ψ 1 (x W g z)(2) 2 plus a constant. The maxmum lkelhood (ML) estmates of the latent mage z and the warpng parameters G are found by computng argmn z,g (x W g z) Ψ 1 (x W g z). 2.2 Warped Illumnaton Subspace The prevous model (1) can only explan the change of appearance from a sngle latent mage. Instead, we want z to be a combnaton of bass mages z = Bs, where B s a D d matrx and s s d-dmensonal vector and d D. Our choce of B comes from the low-dmensonal nature of llumnaton subspaces of convex Lambertan objects: an mage llumnated from arbtrary lght source dstrbuton can be approxmated well by a combnaton of four or nne-dmensonal bass (Basr and Jacobs, 2003). In ths settng, B encodes 3D structure, albedo and the low-pass flterng nature of Lambertan reflectance, and the varables s determne the dstrbuton of lght sources. Let us call s llumnaton coeffcents. The correspondng generatve model s x = W g Bs + ɛ. (3) Snce the warpng W g s a lnear transform of mages, the W g maps a subspace to another subspace. In ths sense we call our model a warped subspace model (refer to Fg. 2). 2.3 Optmzaton Gven an ensemble of mages of the same object wth unknown llumnaton and pose, we can learn g and s by mnmzng the ML cost: C = 2 log p(x B, S, G) = x W g Bs 2 Ψ. (4) We wll use the notaton y 2 Ψ to denote the quadratc form y 2 Ψ = y Ψ 1 y. In superresoluton approaches, the warpng varables g are typcally computed once n a preprocessng step. Snce we are dealng wth mages under arbtrary llumnaton change, a drect regstraton based on ntensty s bound to fal. We overcome ths problem by updatng the regstraton varable g and llumnaton coeffcent s n an alternatng fashon Estmatng Warpng Varables g To mnmze (4) wth respect to g we use a multscale regstraton technque (Vasconcelos and Lppman, 2005) to speed up computatons and avod local mnma. At each level of coarse-tofne mage resolutons, regstraton s done by the Gauss-Newton method descrbed n the followng.

4 Orgnal Bass (B) Illumnated (x=bs) Warped (x=wbs) Fgure 3: Modelng pose and lghtng changes wth a warped subspace model. By estmatng the llumnaton coeffcents s and the warpng varables g wth a known bass B, we can mtate the orgnal pose and lghtng on the left by the lnear combnaton (x = Bs) followed by the geometrcal warpng (x = W gbs). For smplcty assume the cost C s the sum of squared dfferences of two mages I 1 and I 2 C = u,v [I 1 (u, v) I 2 (T 1 g (u, v))] 2. (5) The Gauss-Newton method fnds the mnmum of (5) by the update rule: g n+1 = g n α( 2 gc g n) 1 g C g n, (6) where g C and 2 gc are the gradent and the Hessan of C, respectvely. These are computed from the frst- and the second-order dervatves of T wth respect to g, and the frst- and the secondorder dervatves of mages wth respect to the coordnates (u, v). To apply the technque, we frst generate mage pyramds for I 1 and I 2, we update g by (6) at the coarsest level of the pyramds untl convergence, and repeat the teraton at the fner levels Estmatng Illumnaton Coeffcents s Mnmzng the cost (4) wth respect to s s straghtforward. By settng C s = 2B W Ψ 1 (W Bs x ) = 0 we get the lnear equaton (B W Ψ 1 W B)s = B W Ψ 1 x, (7) whch can be drectly solved by nvertng d d matrx. Note that d D. 2.4 Experments wth Synthetc Data We test the alternatng mnmzaton scheme wth synthetc face data. For ths purpose, we frst generated synthetc mages from a 3D model of a person wth a fxed pose and varyng llumnatons, from whch an emprcal bass B s computed by sngular value decomposton (SVD). The number of bass vector d = 5 was chosen to contan more than 98.8 percent of the total energy, whch agrees wth emprcal (Epsten et al., 1995) and analytcal (Ramamoorth, 2002) studes. A number of mages were randomly rendered wth varyng pose ( yaw 15, ptch 12 ) and varyng lght source drecton ( yaw 60, ptch 50 ). The nose statstcs Ψ were manually determned from the statstcs of the error between the true and the reconstructed mages. Fgure 3 shows the bass mages and the result of jont estmaton. 3 SIMULTANEOUS LEARNING OF BASIS In the prevous secton we showed how to jontly estmate pose and lghtng condtons of test mages from a known bass. However, n practce we do not know the bass for a gven person beforehand. In ths secton we demonstrate an effcent method of learnng the bass as well. 3.1 MAP Estmaton By consderng B also as an unknown varable the ML cost (4) becomes certanly harder to mnmze. The man dffculty les n the degeneracy of the product W Bs. Frst, the product Bs s degenerate up to matrx multplcatons of any nonsngular matrx A, that s, Bs = BA 1 As. We wll mpose orthogonalty on the bass B B = I, but the B and s are stll degenerate up to a d d rotaton matrx. Secondly, W and Bs are also degenerate. One can warp the bass by some transformaton T and compensate t by ts n-

5 verse: W Bs = W T 1 T Bs. To releve the dffculty, we assume a Gaussan process pror on B and a Gaussan densty pror on g. These prors can break the degeneracy by preferrng specfc values of (W, B, s) among those whch gve the same value of W Bs Gaussan Process Pror for Images In super-resoluton problems, fndng latent mages z s usually ll-posed and a pror s requred for z. One of the commonly used prors s the Gaussan MRF (Capel and Zsserman, 2003) of the form p(z) = 1 Z exp( z Qz), whch can be vewed as a Gaussan random process pror on z N (µ, Φ) whose covarance reflects the MRF propertes. We propose the followng pror: f b j denotes the j-th column of B, then each {b j } has the..d. Gaussan pror b j N (µ, Φ). It s typcal to assume µ = 0 for allow arbtrary mages, but we can also use the emprcal bass mages, such as those from the prevous experments wth synthetc faces. We further assume b j s are ndepent. For Φ we choose RBF covarance descrbed n (Tppng and Bshop, 2002): f the - th and j-th entres of z correspond to the pxels (u, v ) and (u j, v j ) n mage coordnates, then [Φ] j exp ( 1 2r {(u 2 u j ) 2 + (v v j ) 2 } ). The nverse Q = Φ 1 of the RBF covarance penalze abrupt changes n the values of nearby pxels and act as a smoothness pror. In practce, the Q = Φ 1 need not be nvertble, and can be made sparse to speed up computatons. The ndependence assumpton on the bass mages {b j } may seem too restrctve, because the bass mages {b j } may be correlated. However, the MRF pror serves manly as the smoothness regularzer and the ndependence only mples that the bass mages are ndependently smooth. Besdes, assumng the full dependency of bass mages s not practcal due to the sheer sze of the covarance matrx Regularzed Warpng A pror for g can prevent unrealstc overregstraton and make the problem better-posed. We penalzed the L 2 norm of the dsplacement feld. Suppose (δu, δv) = T g (u, v) (u, v) s the dsplacement feld of the transform T g, then the norm (u,v) (δu)2 + (δv) 2 measure the overall dstorton the transform T g nduces. The secondorder approxmaton to the squared norm around g = 0 s (u,v) {( g δu) g} 2 + {( g δv) g} 2 = g Λg The correspondng regularzaton term log p(g) = λg Λg s added to the regstraton error term (5) n our multscale regstraton procedure. For perspectve transforms, the Λ s a 8 8 matrx whch doesn t depend on data. 3.2 Optmzaton The proposed MAP cost s as follows: C = 2 log [p(x B, S, G)p(B)p(G)] = x W g Bs 2 Ψ + λ +ηn j g Λg b j µ j 2 Φ + const (8) The mnmzaton s smlarly done by alternatng between mnmzatons over B, s and g. Mnmzng over the latter two s the same as prevous secton, and we only descrbe mnmzaton over B Fndng Bass Images B The dervatve of C w.r.t B s C B = 2 W Ψ 1 (W Bs x )s +2ηNΦ 1 (B µ), where µ = [µ 1...µ d ]. An exact soluton s gven by settng C B = 0: W Ψ 1 (W Bs x )s + ηnφ 1 (B µ) = 0. (9) We can solve the equaton ether drectly or by a conjugate gradent method. After updatng B, we orthogonalze t by a Gram-Schmdt procedure. 3.3 Algorthm The fnal algorthm s summarzed below: 1. ntalze g, s and B. 2. for = 1,..., N, mnmze C over g by multscale regstraton. 3. for = 1,..., N, solve (7) for s by nverson. 4. solve (9) for B by scaled conjugate gradent. 5. orthogonalze B by Gram-Schmdt procedure. 6. repeat 2 5 untl convergence.

6 4 APPLICATIONS TO FACE RECOGNITION In ths secton we perform predcton and recognton experments wth real mages from the Yale face and CMU-PIE databases. 4.1 Yale Face Database The Yale face database (Georghades et al., 2001) conssts of mages from 10 subjects under 9 dfferent poses and 43 dfferent lghtng condtons. 1 From the orgnal mages we roughly crop face regons and resze them to mages (D = 1600). Each mage s then normalzed to have the same sum-of-squares. All mages are globally rescaled to have the mn/max value of 0/1. As a tranng set of each subject we randomly select two mages wth arbtrary lghtng condton per pose, to get a total of 2 9 = 18 mages of unknown pose and llumnaton of the subject. The test set of each subject comprses all remanng mages whch are not n the tranng set ( = 369) of the subject. The 43 lghtng condtons n the test set are dvded nto four subsets as explaned n (Georghades et al., 2001) accordng to the angle the lght source drecton makes wth the frontal drecton. The four subsets conssts of 6, 12, 12, and 13 lghtng condtons respectvely wth ncreasng angles. We generated multple pars of tranng/test sets to get averaged results from random choces of tranng set. 4.2 CMU PIE Database The CMU-PIE database (Sm et al., 2003) conssts of mages from 68 subjects under 13 dfferent poses and 43 dfferent lghtng condtons. We arbtrarly chose the same number (=10) of subjects as the Yale face n our tests.among 13 orgnal camera poses, we have chosen 7 poses whose angles wth the frontal camera pose are roughly less than 40, 2 and 21 lghtng condtons wthout background lghts. We smlarly detect and crop face regons from the orgnal data to a sze of and rescale the ntensty. As a tranng set of each subject we randomly select two mages of arbtrary lghtng condton 1 two of the lghtng condtons ( A-005E+10 and A+050E-40 ) are dropped due to erroneous recordng. 2 cameras numbered 05, 07, 09, 11, 27, 29, 37 per pose to get a total of 2 7 = 14 mages of unknown pose and llumnaton of the subject. The test set of each subject comprses all remanng mages whch are not n the tranng set ( = 133) of the subject. We also dvded the testng set nto three subsets, accordng to the angle the lght source drecton makes wth the frontal axs (0 20, 20 35, ). The four subsets conssts of 7, 6, and 8 lghtng condtons respectvely. We also generated multple pars of tranng/test sets for CMU PIE. 4.3 Parameter Selecton In an earler secton we emprcally determned the value of d and Ψ from synthetc data. Smlarly, we manually chose the parameters {λ, η, Φ} by expermentng wth synthetc faces. The λ relates to the amount of warpng and {η, Φ} relates to the smoothness of mages. As these parameters do not reflect the pecularty of each person, we can the same values for the Yale face and CMU PIE database as the values for synthetc data wthout exhaustve fne-tunngs. 4.4 Predcton Results We demonstrate the advantage of havng a nonlnear warpng to a lnear subspace model n predcton. From 18 (and 14) mages of a person from the Yale face (and CMU PIE) database, we learn the two sets of bases: Lnear subspace: 5-dmensonal bass B j ln s computed from SVD of the tranng data of j-th subject. The predcton ˆx s gven by ˆx = B j ln (Bj ln ) x. Warped subspace: 5-dmensonal bass B j warp, warpng varable g j, and lght coeffcents s j are computed from the tranng data of j-th subject, by mnmzng the MAP cost (8) teratvely from the ntal value B j ln. The predcton ˆx s gven by ˆx = W g j B j warps j. Fgure 4 shows sample results from the two methods. Quanttatve evaluatons are performed as follows. Predcton error for each mage s defned as the fractonal error between true the mage x and the predcted mage ˆx: err = x ˆx 2 / x 2 averaged over all test mages. The result s shown n Table 1. Our method reduces the error to percent of the error from the lnear model.

7 Orgnal Lnear subspace Warped subspace Lnear subspace Warped subspace Learned bass : Orgnal Learned bass : Fgure 4: Experments wth the Yale face (upper) and CMU PIE (lower) databases. Reconstructons of the orgnal mages wth a lnear subspace model (mddle) and wth a warped subspace model (rght) are compared along wth the learned bases from the two methods. The warped subspace model shows ncreased resolutons and sharpness n the learned bass and the reconstructed mages. Subset # Lnear Warped Subset # Lnear Warped Table 1: Average predcton error kx x k2 /kxk2 of lnear vs warped subspace models from the Yale face (upper) and and CMU PIE (lower) databases. 4.5 Recognton Results We compare recognton performance of our method wth four other standard appearancebased methods suggested n (Georghades et al., 2001). These nclude the nearest-neghbor classfers whch do not use subject dentty n tranng: correlaton, egenface, and egenface wthout the frst three egenvectors. For these we have used 50 egenvectors. The other two methods are the same as n the predcton experments (lnear subspace and warped subspace). The average recognton errors were computed over all condtons and all subjects from Yale face and CMU PIE databases, shown n Fg CONCLUSION The Egenface s an almost two-decade old subspace model. Stll, t serves as a fundamental mage-based model due to ts smplcty. In ths work we revamped the subspace model to a warped subspace model whch can cope wth both the lnear varablty n llumnaton and the nonlnear varablty n pose. Gven a few tranng mages, the model can estmate the bass, pose and llumnaton condtons smultaneously va MAP estmaton and multscale regstraton technque. Expermental results confrm the advantage of the warped subspace model over the standard magebased models n predcton and recognton tasks.

8 Yale face CMU PIE Error rate corr eg egwo3 lnear warped Subset # Subset # Fgure 5: Average recognton error of fve algorthms for the Yale face (left) and CMU PIE (rght) databases. The warped subspace model acheves the smallest error across dfferent test subsets of lghtng varatons. We are currently workng on ncreasng the range of pose our model can handle. REFERENCES Basr, R. and Jacobs, D. W. (2003). Lambertan reflectance and lnear subspaces. IEEE Trans. Pattern Analyss and Machne Intellgence, 25(2): Capel, D. and Zsserman, A. (2003). Computer vson appled to super resoluton. IEEE Sgnal Processng Magazne, 20(3): Capel, D. P. and Zsserman, A. (2001). Superresoluton from multple vews usng learnt mage models. In CVPR, volume 2, pages Epsten, R., Hallnan, P., and Yulle, A. (1995). 5 ± 2 Egenmages suffce: An emprcal nvestgaton of low-dmensonal lghtng models. In Proceedngs of IEEE Workshop on Physcs-Based Modelng n Computer Vson, pages Frey, B. J. and Jojc, N. (1999). Transformed component analyss: Jont estmaton of spatal transformatons and mage components. In ICCV, page 1190, Washngton, DC, USA. IEEE Computer Socety. Georghades, A. S., Belhumeur, P. N., and Kregman, D. J. (2001). From few to many: Illumnaton cone models for face recognton under varable lghtng and pose. IEEE Trans. Pattern Analyss and Machne Intellgence, 23(6): Gross, R., Matthews, I., and Baker, S. (2002a). Egen lght-felds and face recognton across pose. In FGR 02: Proceedngs of the Ffth IEEE Internatonal Conference on Automatc Face and Gesture Recognton, page 3, Washngton, DC, USA. IEEE Computer Socety. Gross, R., Matthews, I., and Baker, S. (2002b). Fsher lght-felds for face recognton across pose and llumnaton. In Proceedngs of the 24th DAGM Symposum on Pattern Recognton, pages , London, UK. Sprnger-Verlag. Gunturk, B. K., Batur, A. U., Altunbasak, Y., III, M. H. H., and Mersereau, R. M. (2003). Egenfacedoman super-resoluton for face recognton. IEEE Trans. Image Processng, 12(5): Hallnan, P. (1994). A low-dmensonal representaton of human faces for arbtrary lghtng condtons. In Proc. IEEE Conf. Computer Vson and Pattern Recognton, pages Harde, R. C., Barnard, K. J., and Armstrong, E. E. (1997). Jont MAP regstraton and hghresoluton mage estmaton usng a sequence of undersampled mages. IEEE Trans. Image Processng, 6(12): Ramamoorth, R. (2002). Analytc PCA constructon for theoretcal analyss of lghtng varablty n mages of a Lambertan object. IEEE Trans. Pattern Analyss and Machne Intellgence, 24(10): Sm, T., Baker, S., and Bsat, M. (2003). The CMU pose, llumnaton, and expresson (PIE) database. IEEE Trans. Pattern Analyss and Machne Intellgence, 25(12): Tppng, M. E. and Bshop, C. M. (2002). Bayesan mage super-resoluton. In Becker, S., Thrun, S., and Obermayer, K., edtors, NIPS, pages MIT Press. Turk, M. and Pentland, A. P. (1991). Egenfaces for recognton. Journal of Cogntve Neuroscence, 3(1): Vasconcelos, N. and Lppman, A. (2005). A multresoluton manfold dstance for nvarant mage smlarty. IEEE Trans. Multmeda, 7(1): Zhou, S. K. and Chellappa, R. (2004). Illumnatng lght feld: Image-based face recognton across llumnatons and poses. In FGR, pages IEEE Computer Socety.