Directional Independent Component Analysis with Tensor Representation

Transcription

1 Drecton Independent Component Anyss wth Tensor Representton Le Zhng 1, Qunxue Go 1,2 nd Dvd Zhng 1 1 Bometrc Reserch Center, The Hong ong Poytechnc Unversty, Hong ong, Chn 2 Schoo of Teecommunctons Engneerng, XIDIAN Unversty, X n, Chn cszhng@comp.poyu.edu.h Abstrct Conventon ndependent component nyss (ICA) erns the sttstc ndependences of 2D vrbes from the trnng mges tht re unfoded to vectors. The unfoded vectors, however, me the ICA suffer from the sm smpe sze (SSS) probem tht eds to the dmensonty demm. Ths pper presents nove drecton mutner ICA method to sove those probems by encodng the nput mge or hgh dmenson dt rry s gener tensor. In ddton, the mode- mtrx of the tensor s re-smped nd re-rrnged to form mode- drecton mge to better expot the drecton nformton n trnng. An gorthm ced mode- drecton ICA s then presented for feture extrcton. Compred wth the conventon ICA nd other subspce nyss gorthms, the proposed method cn grety evte the SSS probem, reduce the computton cost n the ernng stge by representng the dt n ower dmenson, nd smutneousy expot the drecton nformton n the hgh dmenson dtset. Experment resuts on we-nown fce nd pmprnt dtbses show tht the proposed method hs hgher recognton ccurcy thn mny exstng ICA, PCA nd even supervsed FLD schemes whe usng ow dmenson of fetures. 1. Introducton How to fnd sutbe representton of the dt s ey probem n pttern nyss, such s fce recognton. Mny subspce nyss methods (SAM) [1-4] hve been proposed to represent the hgh-dmenson dt nto compct ow-dmenson spce to extrct fthfuy the menngfu nd unque structures embedded n the dt. The most representtve unsupervsed SAM technque my be the prncp component nyss (PCA) [1-2]. PCA expots the second-order correton of the trnng dtsets but t gnores the hgher-order sttstc dependences, whch my contn more structur nformton of the 2D or hgher dmenson dt for the subsequent feture cssfcton [3, 4]. Independent component nyss (ICA), s n extenson of PCA, extrcts set of sttstcy ndependent components by nyzng the hgher-order sttstcs n the trnng dtset [5]. Mny schemes hve been reported recenty by usng ICA for fce representton nd recognton [6-9]. These wors cn be genery cssfed nto two groups: one s to study how to evute the performnce of ICA [6-7] nd the other s to study how to mprove the performnce of ICA n feture extrcton nd cssfcton [8-9]. In those gorthms, ner gebr s used to extrct the feture of ndependent components (IC). Thus they re hrd to dstngush the sttstc fetures rse from dfferent fctors, or modes, nherent to mge formton, such s vewpont, umnton, etc., [4]. To overcome ths probem of dfferent mgng fctors, recenty Vsescu et [4] used mutner gebr to represent the dtsets nd extrct ICs nd they obtned better performnce. However, the bove gorthms stretch the nput mge nto vector for IC extrcton. The unfoded vector my ose some structur nformton embedded n mge nd w ed to very hgh dmensonty of the dt for the subsequent nyss. The vbe number of trnng smpes s usuy much smer thn the dmensonty of unfoded vector n prctc ppctons. Ths s the so ced sm smpe sze (SSS) probem n SAM bsed pttern recognton. Some recent wors hve been tng the mge drecty s two dmenson mtrx or hgh-order tensor for sttstc ernng nd hve obtned good resuts n bometrc uthentcton [10-13]. However, few wors hve been reported for two dmenson mtrx or tensor bsed ICA. Athough tensor representton s used n [4], the nput mge s st unfoded to vector before ppyng ICA. In ths pper, we nvestgte how to mpement ICA by encodng the mge s 2 nd (or hgher order) tensor, nd propose frmewor for IC extrcton by usng drecton tensor mge representton. The proposed method uses mutpe nterreted subspces correspondng to dfferent tensor dmensons rther thn one subspce s n trdton ICA for IC extrcton. An effcent ernng procedure s presented v nove tensor nyss, ced mode /08/$ IEEE 1 Authorzed censed use mted to: Hong ong Poytechnc Unversty. Downoded on September 10, 2009 t 03:40 from IEEE Xpore. Restrctons ppy.

2 drecton ICA. Dfferent from trdton tensor nyss tht drecty erns mode- subspce from the mode- tensor mges, the proposed mode- drecton ICA erns the ow-dmenson subspce from the mode- drecton mges, whch re formed by re-smpng nd re-rrngng the mode- mtrx of the orgn tensor. The mode- drecton mges re vewed s the new subjects to be nyzed n the th subspce. Compred wth conventon ICA gorthms, the proposed method evtes grety the SSS probem nd hence the dmensonty demm. In the proposed mode- drecton ICA, the dmensonty of vrbes s reduced to the th dmenson of the tensor mge, whe the smpe sze s ncresed by rge fctor. On the other hnd, more usefu structur nformton embedded n trnng mges s preserved nd the drecton nformton cn so be embedded n the tensor representton. Experments on UMIST nd AR fce dtbses nd the pmprnt dtbse show tht the proposed method cheves hgher recognton ccurcy whe usng ower dmenson of fetures. The remnder of ths pper s orgnzed s foows. Bcground nd nottons of ICA nd mutner gebr re presented n secton 2. The mutner drecton ICA gorthm s presented n Secton 3. Secton 4 presents extensve experments nd Secton 5 concudes the pper. 2. Bcground nd Nottons 2.1. Independent component nyss (ICA) In [3], Brtett et proposed two rchtectures for ICA. Here we use the rchtecture I. Denote by x p-dmenson mge vector, the ICA of x sees for sequence of projecton vectors w 1, w 2,, w (q<p) to mxmze the q sttstc ndependence of the projected dt. It cn be expressed s foows: T s W x (1) where s denotes the ICs of x nd W w,, w 1 q s ced the projecton mtrx. Vrous crter, such s those bsed on mutu nformton, negentropy nd hgher-order cumunts, hve been proposed for computng W [5]. Among them the FstICA gorthm hs been wdey used n pttern recognton [5, 9]. Usuy, PCA s mpemented to whten the dt nd reduce the dmensonty before ppyng ICA. Ntur mges re usuy represented n the form of mtrces (2 nd order tensor) or hgher-order tensors. Therefore t s not we suted to represent ntur mges usng one-dmenson vectors. The mge-to-vector trnsformton so eds to the SSS probem nd the dmensonty demm. To ddress these probems n conventon ICA, we w propose nove mutner drecton ICA scheme n Secton Mutner gebr Ths secton brefy ntroduces the concepts nd nottons of mutner gebr [4, 14] tht w be used n the foowng deveopment. A tensor s hgher order generzton of vector (1 st order tensor) nd mtrx (2 nd order tensor) nd t s mutner mppng over set of vector spces. Denote by I1I2I A tensor of order. The sze of the th dmenson of A s I. An eement of A s denoted s A or 1 1, where 1 I. In tensor termnoogy, mtrx coumn nd row vectors re referred to s mode-1 nd mode-2 vectors, respectvey. For hgher-order tensors, we hve the foowng defntons. Defnton 1. [14] (Mode- mtrxzng or mtrx unfodng) The mode- mtrxzng or mtrx unfodng of th I I order tensor A s mtrx D, I j j I. I D s the ensembe of vectors n obtned by eepng ndex fxed nd vryng the other ndces. Defnton 2. [14] (Mode- product) The mode- product I I' A U of tensor A nd mtrx U s ' I1I2I 1I I 1 I tensor defned by A U A U (2) j 121j for ndex vues. The mode- product s type of contrcton. Defnton 3. [4, 14] (Mode- vectors) The mode- vectors of th order tensor A re the I -dmenson vectors obtned from A by vryng ndex whe eepng the other ndces fxed. The mode- vectors re the coumn I vectors of mtrx I1I 1I 1I D resuted by mode- mtrxzng the tensor A. Defnton 4. [14] (-rn) The -rn of tensor I1I2I A, denoted by R, s defned s the dmenson of the vector spce generzed by the mode- vectors. Defnton 5. [4] The dstnce between tensors A nd B s defned s d A,B A -B,A -B (3) where I1,, I A,B A B 1 1,, denotes the nner product of tensors A nd B wth the sme dmenson. 2 Authorzed censed use mted to: Hong ong Poytechnc Unversty. Downoded on September 10, 2009 t 03:40 from IEEE Xpore. Restrctons ppy.

3 3. Drecton Tensor ICA 3.1. Mode of ICA wth tensor representton Most exstng ICA gorthms consder n mge s vector nd thus hve very hgh dmenson of feture spce. As resut, these methods suffer from the SSS probem due to the dmensonty demm. Imges cn be more ntury represented s 2 nd or hgher order tensors. In ths secton, we study how to perform ICA wth gener tensor representton of mges. Gven n rbtrry th I1 I2 I order tensor A, t cn be expressed s foows [14] S A 1U1 2U2 3 U (4) where tensor S, ced the core tensor, governs the LI ntercton between the mtrces U ( L I, =1,2,,), whch re ced ower-dmenson ndependent subspces. Mtrx U contns the orthogon vectors spnnng the coumn spce of the mtrx D tht ( ) resuted from the mode- mtrxzng of A. Our go s to fnd trnsformton mtrces U such tht the eements of S re s ndependent s possbe. The proposed mode my seem smr to the mode proposed by Vsescu [4]. However, they re very dfferent n essence. In mode (4), A denotes hgh dmenson dtset (e.g. n mge) tht s represented n tensor form rther thn vector s n [4]. It cn thus evte sgnfcnty the SSS probem n SAM. The mtrces U n the proposed mode cn be used to represent the mode- dmenson of tensor mges wth fctors such s umnton, pose, etc. In ths pper, we w propose frmewor to estmte mutpe subspces U n (4) by obtnng the mode- drecton mges from tensor A nd then vewng the mode- vectors of the drecton mges s trnng smpes. The method w be descrbed detedy n the foowng sub-sectons Mode- drecton mge In subspce nyss wth tensor representton, the th subspce, contnng the th dmenson structur nformton of the tensor, cn be drecty ccuted from the mode- mtrx obtned by unfodng the tensor usng Defnton 1. However, the mode- mtrxzng gnores the retonshp between the current dmenson nd other dmensons, whch s reted to the mge formton process nd my be usefu for cssfcton. To sove ths probem nd mprove the cssfcton ccurcy, we propose to use mode- drecton mges to estmte the th subspce. The mode- drecton mge s defned nd obtned from the orgn tensor s foows. Defnton 6. (Mode- drecton mge) Gven th order I1I2I tensor A, the mode- (=1,2,,) drecton mge s obtned s foows: I I.) Obtn the mode- mtrx D ( ) from tensor A v Defnton 1..) Re-smpe nd re-rrnge the mode- mtrx D ( ) ong the I -dmenson drecton to generte the mode- drecton mge B, s shown n Fg. 1. The nteger prmeter contros the resuted drecton mge. Defnton 7. (Mode- drecton vectors) The mode- drecton vectors of th order tensor A re the I -dmenson vectors of drecton mge B,.e., mode-2 vectors of B. When s equ to I, the mode- drecton mge B s the mode- mtrx D ( ) nd the re-smpng drecton s of zero degree. When s 1, the drecton mge B w be the dgon mge of D. In Fg. 2, we ( ) show n exmpes when s 2 nd 4 for the 2 nd order tensor,.e. two dmenson mge. The mode-1 nd mode-2 drecton mges of the orgn mge (Fg. 2 ()) re shown n Fgs. 2 (b), (c), (d) nd (e), respectvey, where the vue of s 2 n (b) nd (c), nd s 4 n (d) nd (e). By constructng the mode- drecton mge, more drecton nformton of the orgn tensor cn be embedded. Compred wth the mode- mtrx D ( ), mode- drecton mge B w be be to empoy the pxes ong the I -dmenson drecton for trnng nd feture extrcton. Next n secton 3.3 we ntroduce the mode- drecton ICA nd then n secton 3.4 we present the drecton tensor ICA gorthm Mode- drecton ICA Before ntroducng the drecton tensor ICA, we frst summrze the mode- drecton ICA of mge A s foows:.) For =1, 2,,, compute the mode- mtrxzng mtrx D ccordng to Defnton 1. ( ).) Form the mode- drecton mge B ccordng to Defnton 6..) Te the mode- drecton vectors (refer to Defnton 7) of A s trnng smpes, nd compute the mtrx U n (4) by usng the FstICA gorthm. 3 Authorzed censed use mted to: Hong ong Poytechnc Unversty. Downoded on September 10, 2009 t 03:40 from IEEE Xpore. Restrctons ppy.

4 Dmensonty reducton n the ner cse does not hve trv mutner counterprt. Accordng to [4, 14], usefu generzton to tensor nvoves n optm rn pproxmton whch tertvey optmzes ech of the modes of the gven tensor. Ech optmzton step w nvove best reduced-rn pproxmton of postve sem-defnte symmetrc mtrx. Ths s hgh-order extenson of the orthogon terton for mtrces. The proposed mode- drecton ICA gorthm vods the tertve step n trnng Drecton tensor ICA gorthm Wth the bove deveopment, the mutner drecton ICA gorthm cn be summrzed s foows: I1I2I.) Input the orgn trnng dtset A, =1,2,, N, where N s the number of trnng smpes nd s the order of tensor A. Set the dmensonty 12 of the output tensor S..) Trnng stge For =1, 2,, Ccute D( ) A usng Defnton 1; Form the drecton mge B for ( ) D by usng ( ) Defnton 6; Ccute the mtrx U for B usng the ( ) mode- ICA gorthm n secton 3.3. End.) Extrct the ICs s foows: S A U U, 1, 2,, N 1 1 * v.) Extrct ICs of the probe mge A : * * S A 1U1 U v.) Cssfcton bsed on the dstnce (refer to Defnton * 5) between S nd S. Note tht the dscrmnbty of ech coumn of U s unnown n pror. In the experments, we used the method proposed n [3] to rerrnge the coumn of U to mprove the cssfcton ccurcy nd reduce the dmensonty of fetures Dscussons In the whtenng stge of conventon ICA, the sze of the covrnce mtrx w be I I f we unfod 1 1 I1 I2 I tensor A to vector. Usuy the trnng smpe sze N I n most prctc ppctons. It s 1 hrd to ccute ccurtey nd robusty the sttstcs of the vector vrbe becuse the trnng smpe sze s much smer thn the dmensonty of the vector vrbe. In the proposed method, however, the sze of the step-wse covrnce mtrx s I I, whch s much smer thn tht of ICA. On the other hnd, s descrbed n secton 3.3, the trnng smpes re the mode- drecton vectors of A nd the number of them s I N, whch s much rger thn I. Therefore, the dmensonty demm s sgnfcnty evted. For th order tensor, there re projecton mtrces ccuted n the proposed gorthm, whch contn the structur nformton embedded n dfferent tensor dmensons. The proposed mutner ICA cn conduct dmensonty reducton from dfferent drectons nd extrct effectvey the drecton fetures. In the trnng process, we form the drecton mge for the mode- mtrx to embed the drecton nformton ong the drecton wth I -dmenson. As we cn see n the experments, ths nove processng cn cheve hgher recognton ccurcy. 4. Experment Resuts In ths secton, we verfy the performnce of the proposed method on pmprnt dtbse nd two benchmr fce dtbses, UMIST [15] nd AR [16]. The proposed method s compred wth both unsupervsed methods, ncudng PCA (Egenfces) [2], ICA [3], 2DPCA [10] nd B-2DPCA [17], nd supervsed methods, ncudng FLD [18], 2DFLD [19] nd tensor-fld [11]. In the experments, we consder the mge s 2 nd order tensor (.e. =2) nd used the nerest neghborhood cssfer for cssfcton Pmprnt dtbse The used pmprnt dtbse ( edu.h/~bometrcs/) ws coected from 50 peope t dfferent tmes. The pmprnts from rght-hnd nd eft-hnd of ech person re treted s pmprnts from dfferent peope. The resouton of the orgn pmprnt mges s After preprocessng, the centr prt of the mge, whose sze s , s cropped for feture extrcton nd mtchng. Fg. 3 shows n exmpe of the preprocessng resut. In the experment, we seected pmprnt mges from 100 dfferent pms for gery wth ech pm hvng 6 smpes ten n two sessons. The smpes from the frst sesson were used for trnng, nd the smpes from the second sesson were used for testng. Thus, the tot number of trnng smpes nd test mges re both 300. Tbe 1 shows the top recognton ccurcy of dfferent schemes wth the correspondng dmensonty of fetures. It cn be seen tht the proposed method s obvousy superor to the other unsupervsed gorthms (PCA, j j 4 Authorzed censed use mted to: Hong ong Poytechnc Unversty. Downoded on September 10, 2009 t 03:40 from IEEE Xpore. Restrctons ppy.

5 2DPCA, B-2DPCA nd ICA) nd even the supervsed methods (FLD 2DFLD nd Tensor-FLD) n recognton ccurcy. However, the proposed method my need more fetures thn conventon ICA. It cn so be seen tht wth sutbe seecton of prmeter (such s =2, 4, 8) n the formton of the mode- drecton mge, the recognton ccurcy w be hgher thn tht wth the orgn mode- mge,.e. when =128. Emprcy we found tht n most experments on pmprnt nd fce dtbses, the hghest cssfcton ccurcy cn be cheved when s UMIST fce dtbse The UMIST dtbse [15] s mut-vew fce dtbse, consstng of 575 mges from 20 peope nd coverng wde rnge of poses from profe to front vews. Fg. 4 shows some mges of one subject. Ech mge s of sze In the experments, the frst nneteen smpes of ech person were used. Then we used the frst p=1, 3, 6, 9 mges for trnng nd used the remnng mges for testng. Tbe 2 sts the top cssfcton ccurces of dfferent gorthms nd the ssocted number of fetures. We see cery tht the proposed method cheves much hgher ccurcy thn the unsupervsed methods PCA, ICA nd 2DPCA, nd even hgher ccurcy thn the supervsed methods FLD, 2DFLD nd Tensor-FLD. The proposed method hs the best cssfcton ccurcy when s AR fce dtbse In the AR dtbse [16], the mges of 120 ndvdus (65 men nd 55 women) were ten n two sessons (seprted by two wees) nd ech sesson contns 13 mges. In our experments, the fc porton of ech mge s mnuy cropped nd then normzed to sze of The mges from the frst sesson wth () neutr expresson, (b) sme, (c) nger, (d) screm, (e) eft ght on, (f) rght ght on, nd (g) both sde ght on were seected for gery. Thus we hve 840 mges from 120 ndvdus. Fg. 5 shows some smpe mges of one subject. Two experments were performed. In the frst experment, the four smpe mges per person wth () neur expresson, (b) sme, (c) nger nd (d) screm n the frst sesson were seected for trnng, nd the other three mges for testng. The second experment exchnges the trnng nd testng mges. Tbe 3 sts the top cssfcton ccurces of dfferent gorthms nd the ssocted number of fetures. We cn hve the sme concuson s n the prevous experments. Fg. 6 pots the recognton ccurcy of the proposed method under dfferent number of fetures n the frst experment. It cn be seen tht recognton ccurcy of the proposed method w ncrese when the number of fetures ncrese, when the number of fetures s 86, t hs the best ccurcy (99.72%). 5. Concuson A gener frmewor for ndependent feture extrcton wth tensor representton ws proposed n ths pper. The proposed method erns mutpe ow-dmenson subspces to extrct ndependent fetures. A nove mode- drecton mge formton ws used n trnng to better expot the drecton nformton n the mode- mtrx of the tensor. Then the mode- ICA ws presented to ern the subspces v mode- drecton mges nd fny the mutner drecton ICA gorthm ws presented to extrct the IC fetures, whch s so n tensor form. Compred wth the trdton ICA gorthms, the proposed method evtes sgnfcnty the sm smpe sze probem nd cn preserve better the structur nformton embedded n the tensor dtsets. From the experments on one pmprnt dtbse nd two fce (UMIST nd AR) dtbses, t cn be concuded tht the proposed gorthm hs hgher cssfcton ccurcy thn mny exstng unsupervsed gorthms such s PCA, ICA nd 2DPCA, nd even supervsed gorthms such s FLD, 2DFLD nd Tensor-FLD. () (b) (c) (d) (e) Fgure 2. Orgn mge nd trnsformed drecton mges. () Orgn 2 nd order tensor mge; (b) mode-1 drecton mge wth =2; (c) mode-2 drecton mge wth =2; (d) mode-1 drecton mge wth =4; (e) mode-2 drecton mge wth =4. 5 Authorzed censed use mted to: Hong ong Poytechnc Unversty. Downoded on September 10, 2009 t 03:40 from IEEE Xpore. Restrctons ppy.

6 Tbe 1. Top recognton ccurces (%) nd the ssocted dmensontes on the Pmprnt dtbse by dfferent schemes. Proposed method Method PCA ICA FLD 2DPCA B-2DPCA Tensor-FLD 2DFLD =2 =1 =4 =8 =128 Accurcy Dmenson Tbe 2. The recognton ccurces (%) of dfferent schemes on the UMIST dtbse. The vues n prentheses re the correspondng number of fetures. Trnng number PCA ICA 2DPCA B-2DPCA FLD Tensor-FLD 2DFLD (18) (16) (46) (48) (17) (18) (50) (25) (672) (336) (336) (336) (60) (30) (42) (14) (8) (10) (9) (140) (72) (48) (224) (448) (224) Proposed method =2 =4 = (27) (20) (36) (16) (27) (33) (14) (18) (22) (20) (22) (24) Tbe 3. The recognton ccurces (%) of dfferent schemes on the AR dtbse. The vues n prentheses re the correspondng number of fetures. Proposed method Method PCA ICA 2DPCA B-2DPCA Tensor-FLD FLD 2DFLD =2 =5 =10 = Frst experment Second experment (418) (110) (99) (96) (1300) (1000) (375) (360) (270) (60) (113) (79) (1150) (250) (108) (72) (91) (99) (96) (88) (91) (88) 1,1... 1, 2,1... 2, 3,1... 3,... n,1 n, 1,1... 1, 2,1... 2, 3,1... 3,... n,1 n,... 1, 1 1, , 1 2,2 3, 1 3,2 n, 1 n,2... 1, 1 1, , 1 2,2 3, 1 3,2 n, 1 n, ,( m1) 1 1, m 2,( m1) 1 2, m 3,( m1) 1 3, m... n,( m1) 1 n, m ,( m1) 1 1, m 2,( m1) 1 2, m... n,( m1) 1 n, m orgn mge 1,1... 1, 2,1... 2, 3,1... 3,... n,1 n, orgn mge drecton mge , 1 2,2 3, 1 3,2 n, 1 n,2 1, 1 1, n,( m1) 1 n, m 1,( m1) 1 1, m 2,( m1) 1 2, m Fgure 1. Mode- drecton mge formton. Fgure 3. Some preprocessed mges (128128) n the pmprnt dtbse. 6 Authorzed censed use mted to: Hong ong Poytechnc Unversty. Downoded on September 10, 2009 t 03:40 from IEEE Xpore. Restrctons ppy.

7 Fgure 4. Some smpe mges of one subject n the UMIST dtbse. Fgure 5. Some smpe mges of one subject n the AR dtbse. Recognton ccurcy Number of fetures Fgure 6. Recognton ccurcy vs. number of fetures by the proposed method (=10) on AR dtbse. References [1] S. Yn, D. Xu, B. Zhng, H. Zhng, Q. Yng nd S. Ln, Grph Embeddng nd Extensons: A Gener Frmewor for Dmensonty Reducton, IEEE Trns. Pttern An. Mch. Inte, 29(1):40-51, [2] M. Tur nd A. Pentnd. Egenfces for recognton. Journ of Cogntve Neuroscence, 3(1):72-86, [3] M. S. Brtett, J. R. Moven nd T. J. Sejnows. Fce recognton by ndependent component nyss. IEEE Trns. on Neur Networs, 13(6): , [4] M. Aex O. Vsescu nd D. Terzopouos. Mutner Independent Component Anyss. In Proc. Of IEEE Conference on Computer Vson nd Pttern Recognton (CVPR05), [5] A. Hyvärnen, J. rhunen, nd E. Oj. Independent component nyss. Wey, New Yor, [6] B. Moghddm. Prncp mnfods nd probbstc subspces for vsu recognton. IEEE Trns. IEEE Trns. Pttern An. Mch. Inte, 24(6): , [7] M. A. Vcente, P. O. Hoyer, nd A. Hyvärnen. Equvence of some common ner feture extrcton technques for ppernce-bsed object recognton tss. IEEE Trns. Pttern An. Mch. Inte, 29(5): , [8] M. Bressn nd J. Vtrà. On the seecton nd cssfcton of ndependent fetures. IEEE Trns. Pttern An. Mch. Inte, 25(10): , [9] J. m, J. M. Cho, J. Y, M. Tur. Effectve representton usng ICA for fce recognton robust to oc dstorton nd prt occuson. IEEE Trns. Pttern An. Mch. Inte. 27(12): , [10] J. Yng, D. Zhng, A. F. Frng, nd J. Y. Yng. Two-dmenson PCA: new pproch to ppernce-bsed fce representton nd recognton. IEEE Trns. Pttern An. Mch. Inte, 26(1): , [11] D. To, X. L, X. Wu. Gener Tensor Dscrmnnt Anyss nd Gbor Fetures for Gt Recognton. IEEE Trns. Pttern An. Mch. Inte. 29(10): , [12] S. Yn, D. X, Q, Yng, L. Zhng, X. Tng nd H. Zhng. Dscrmnnt nyss wth tensor representton. In proc. Of IEEE conference on Computer Vson nd Pttern Recognton (CVPR), [13] T. m, S. Wong, nd R. Cpo. Tensor cnonc correton nyss for cton cssfcton. In proc. Of IEEE conference on Computer Vson nd Pttern Recognton (CVPR), [14] L. Lthuwer, B. Moor, nd J. Vndewe. A mutner sngur vue decomposton. SIAM J. Mtrx An. App., 21(4): , [15] UMIST Fce Dtbse, Dne Grhm, ee.umst.c.u/dnny/dtbse.htm, [16] The AR Fce Dtbse, ~ex/ex _fce_db.htm. [17] J.Ye. Generzed ow rn pproxmtons of mtrces. Mchne Lernng, 61: , [18] P. N. Behumeur, J. P. Hespnh, nd D. J. regmn. Egenfces vs. Fsherfces: recognton usng css specfc ner projecton. IEEE Trns. Pttern An. Mch. Inte, 19(7): , [19]. Lu, Y. Cheng nd J. Yng. Agebrc feture extrcton for mge recognton bsed on n optm dscrmnnt crteron. Pttern Recognton, 26(6): , Authorzed censed use mted to: Hong ong Poytechnc Unversty. Downoded on September 10, 2009 t 03:40 from IEEE Xpore. Restrctons ppy.