Weighted Piecewise LDA for Solving the Small Sample Size Problem in Face Verification

Transcription

1 TNN05-P485 Weghted Pecewse LDA fo Solvng the Small Sample Sze Poblem n Face Vefcaton Maos Kypeountas, Anastasos Tefas, Membe, IEEE, and Ioanns Ptas, Seno Membe, IEEE Abstact A novel algothm that can be used to boost the pefomance of face vefcaton methods that utlze Fshe s cteon s pesented and evaluated. The algothm s appled to smlaty, o matchng eo, data and povdes a geneal soluton fo ovecomng the small sample sze (SSS poblem, whee the lack of suffcent tanng samples causes mpope estmaton of a lnea sepaaton hype-plane between the classes. Two ndependent phases consttute the poposed method. Intally, a set of weghted pecewse dscmnant hype-planes ae used n ode to povde a moe accuate dscmnant decson than the one poduced by the tadtonal lnea dscmnant analyss (LDA methodology. The expected classfcaton ablty of ths method s nvestgated thoughout a sees of smulatons. The second phase defnes pope combnatons fo peson-specfc smlaty scoes and descbes an outle emoval pocess that futhe enhances the classfcaton ablty. The poposed technque has been tested on the M2VTS and XM2VTS fontal face databases. Expemental esults ndcate that the poposed famewok geatly mpoves the face vefcaton pefomance. Index Tems Face vefcaton, lnea dscmnant analyss (LDA, small sample sze (SSS poblem L I. INTRODUCTION INEAR dscmnant analyss s an mpotant statstcal tool fo patten ecognton, vefcaton, and, n geneal, classfcaton applcatons. It has been shown that LDA can be effectve n face ecognton o vefcaton poblems [, 2, 3]. In face ecognton systems, the Ν closest faces, fom a set of efeence faces, to a test face ae found. In face vefcaton systems, a test face s compaed aganst a efeence face and a decson s made whethe the test face s dentcal to the efeence face (meanng the test face coesponds to a clent o not (meanng the test face coesponds to an mposto. The afoementoned poblems ae conceptually dffeent. On one hand, a face ecognton system usually asssts a human faceecognton expet to detemne the dentty of the test face by Manuscpt eceved Apl 5, Ths wok was patally funded by the ntegated poject BoSec IST (Bometc Secuty, unde Infomaton Socety Technologes (IST poty of the 6th Famewok Pogamme of the Euopean Communty and s patally funded by the netwok of excellence BoSecue IST (Bometcs fo Secue Authentcaton, unde Infomaton Socety Technologes (IST poty of the 6th Famewok Pogamme of the Euopean Communty. The authos ae wth the Depatment of Infomatcs, Astotle Unvesty of Thessalonk, Thessalonk 54006, Geece (e-mal: ptas@aa.csd.auth.g. computng all smlaty scoes between the test face and each human face stoed n the system database and by ankng them. On the othe hand, a face vefcaton system should decde tself f the test face s a clent o an mposto [4]. The evaluaton ctea fo face ecognton systems ae dffeent fom those appled to face vefcaton systems. The pefomance of face ecognton systems s quantfed n tems of the pecentage of coectly dentfed faces wthn the Ν best matches. By vayng the ank Ν of the match, the cuve of cumulatve match scoe vesus ank s obtaned [5]. The pefomance of face vefcaton systems s measued n tems of the false ejecton ate (FRR acheved at a fxed false acceptance ate (FAR o vce vesa. By vayng FAR, the Receve Opeatng Chaactestc (ROC cuve s obtaned. Fo a face vefcaton system, thee s a tade-off between the FAR and the FRR. The choce of the pefomance metc,.e., FAR o FRR, that should be low depends on the natue of the applcaton [6]. If a scala fgue of met s used to judge the pefomance of a vefcaton algothm, t s usually the opeatng pont whee the FAR and FRR ae equal, the so called Equal Eo Rate (EER. A thd dffeence s n the equements needed when face ecognton/vefcaton systems ae taned. Face ecognton systems ae usually taned on sets havng one fontal mage pe peson. Fo example, n face ecognton expements conducted on FERET database [7], the fa (egula facal expesson fontal mages ae used to tan the system, whle the fb (altenatve facal expesson fontal mages ae used to test the system. Face vefcaton systems usually need moe mages pe ndvdual fo tanng to captue nta-class vaablty (.e., to model the vaatons of the face mages coespondng to the same ndvdual. The equements n the numbe of mages ncease damatcally when lnea dscmnant analyss s employed to accomplsh featue selecton [8]. In many cases, the avalable facal mages ae nsuffcent fo cayng out the LDA pocess n a statstcally pope manne. In ths type of poblems, Fshe s lnea dscmnant [9] s not expected to be able to dscmnate well between face patten dstbutons that n many cases cannot be sepaated lnealy, unless a suffcently lage tanng set s avalable. Moe specfcally, n face ecognton o vefcaton systems LDA-based appoaches often suffe fom the SSS poblem, whee the sample dmensonalty s lage than the numbe of avalable tanng samples pe subject [0]. In fact, when ths poblem becomes sevee, tadtonal LDA shows poo genealzaton ablty and degades the classfcaton pefomance.

2 TNN05-P485 2 In ecent yeas, an nceasng nteest has developed n the eseach communty n ode to mpove LDA-based methods and povde solutons fo the SSS poblem. The tadtonal soluton to ths poblem s to apply LDA n a lowedmensonal PCA subspace, so as to dscad the null space (.e., the subspace defned by the egenvectos that coespond to zeo egenvalues of the wthn-class scatte matx of the tanng data set []. Howeve, t has been shown [] that sgnfcant dscmnant nfomaton s contaned n the dscaded space and altenatve solutons have been sought. Specfcally, n [2] a dect-lda algothm s pesented that dscads the null space of the between-class scatte matx, whch s clamed to contan no useful nfomaton, athe than dscad the null space of the wthn-class scatte matx. Ths appoach was also used n [3], whee a subspace of the null space of the wthn-class scatte matx s used to solve the small sample sze poblem. Fst the common null space of the between-class scatte matx and the wthn-class scatte matx s emoved, snce t s useless fo dscmnaton. Then, the null space of the esultng wthn-class scatte matx s calculated n the lowe-dmensonal pojected space. Ths null space, combned wth the pevous pojecton, epesents a subspace whch s useful fo dscmnaton. The optmal dscmnant vectos of LDA ae deved fom t. The key to the appoach n [4] s to use the dect-lda technques fo dmensonalty educton and meanwhle utlze a modfed Fshe cteon that s moe closely elated to the classfcaton eo. To obtan ths modfed cteon, weghted schemes should be ntoduced nto the tadtonal Fshe cteon to penalze the classes that ae close and can lead to potental msclassfcatons n the output space. In [4], howeve, smple weghted schemes ae ntoduced nto the econstucton of the between-class scatte matx n the dmensonalty educed subspace, such that the optmzaton can be caed out by solvng a genealzed egenvalue poblem wthout havng to esot to complex teatve optmzaton schemes. The method n [5] utlzes a vaant of dect-lda to safely emove the null space of the betweenclass scatte matx and apples a factonal step LDA scheme to enhance the dscmnatoy powe of the obtaned dect- LDA featue space. Moe ecently, the authos n [0] fomed a mxtue of LDA models that can be used to addess the hgh nonlneaty n face patten dstbutons, a poblem that s commonly encounteed n complex face ecognton tasks. They pesent a machne-leanng technque that s able to boost an ensemble of weak leanes, opeatng slghtly bette than andom guessng, to a moe accuate leane. In [6], a lnea featue extacton method whch s capable of devng dscmnatoy nfomaton of the LDA cteon n sngula cases s used. Ths s a two-stage method, whee PCA s fst used to educe the dmensonalty of the ognal space and then a Fshe-based lnea algothm, called Optmal Fshe Lnea Dscmnant, fnds the best lnea dscmnant featues on the PCA subspace. One of the majo dsadvantages of usng the Fshe cteon s that the numbe of ts dscmnatng vectos capable to be found s equal to the numbe of classes mnus one. Recently, t was shown [7] that altenatve LDA schemes that gve moe than one dscmnatve dmensons, n a two class poblem, have bette classfcaton pefomance than those that gve one pojecton. Ths s done by only eplacng the ognal between scatte wth a new scatte measue. In anothe attempt to addess the SSS poblem, the authos n [8] pesent the egulazed LDA method (RLDA that employs a egulazed Fshe s sepaablty cteon. The pupose of egulazaton s to educe the hgh vaance elated to the egenvalue estmates of the wthn-class scatte matx, at the expense of potentally nceased bas. By adjustng the egulazaton paamete R, a set of LDA vaants ae obtaned, such as the dect-lda of [2] fo R = 0, and the DLDA of [5] fo R =. The tade-off between the vaance and the bas, dependng on the sevety of the SSS poblem, s contolled by the stength of egulazaton. The detemnaton of the optmal value fo R s computatonally demandng as t s based on exhaustve seach [8]. Smlaly, n [9] a new Quadatc Dscmnant Analyss (QDA-lke method that effectvely addesses the SSS poblem usng a egulazaton technque s pesented. The dect-lda technque s utlzed to map the ognal face pattens to a low-dmensonal dscmnant featue space, whee a egulazed QDA s then eadly appled. The egulazaton stategy used povdes a balance between the vaance and the bas n sample-based estmates and ths sgnfcantly eleves the SSS poblem. In [20] a kenel machne-based dscmnant analyss method, whch deals wth the nonlneaty of the face pattens dstbuton s poposed whch also attempts to solve the SSS poblem. Intally, the ognal nput space s non-lnealy mapped to an mplct hgh-dmensonal featue space, whee the dstbuton of face pattens s hoped to be lneazed and smplfed. Then, a new vaant of the dect-lda method s ntoduced to effectvely solve the SSS poblem and deve a set of optmal dscmnant bass vectos n the featue space. Unlke the ognal dect-lda method of [2], zeo egenvalues of the wthn-class scatte matx ae neve used as dvsos n the poposed one. In ths way, the optmal dscmnant featues can be exactly extacted fom both nsde and outsde of the wthn-class scatte matx s null space. In [2] the kenel tck s appled to tansfom the lnea-doman Foley-Sammon optmal w..t. othogonalty constants dscmnant vectos, esultng n a new nonlnea featue extacton method. The Foley-Sammon method can obtan moe dscmnant vectos than LDA, howeve, t does not show good pefomance when havng to deal wth nonlnea pattens, such as face pattens. Thus, the kenel tck s employed to povde a nonlnea soluton. In addton, ths method handles the SSS poblem effectvely by ensung that most of ts dscmnant solutons le n the null space of the wthn-class matx. In [22] a kenel optmzaton method s pesented that maxmzes a measue of class sepaablty n the empcal featue space. The empcal featue space s a

3 TNN05-P485 3 Eucldean space n whch the tanng data ae embedded n such a way that the geometcal stuctue such as pa-wse dstance and angle n ths featue space s peseved. Ths leads to a data-dependent kenel optmzaton capablty whee the optmzed kenel can mpove classfcaton pefomance. The featue selecton va lnea pogammng (FSLP method [23] ncopoates a featue selecton pocess based on magn sze, whee magn s defned as the mnmum dstance between two boundng hype-planes. The FSLP method can select featues by maxmzng the magn, thus ccumventng the cuse of dmensonalty poblem n the small sample case, when the numbe of featues s lage. In addton, pawse featue selecton s employed to choose the most elevant featues fo each pa of classes athe than select a fxed subset of featues fo each class to dscmnate t fom all othe classes. The FSLP technque detemnes the numbe of featues to select fo each pa of classes. In [24] a pobablstc model s used to genealze LDA n fndng components that ae nfomatve of o elevant fo data classes, thus emovng the estctve assumpton of nomal dstbuton wth equal covaance matces n each class. The dscmnatve components maxmze the pedctablty of the class dstbuton whch s asymptotcally equvalent to maxmzng mutual nfomaton wthn the classes and fndng pncpal components n the so-called leanng o Fshe metcs. In [25] vefcaton pefomance was nceased by employng a dvde-and-conque technque. Specfcally, a suppot vecto machne (SVM tee s used, n whch the sze of the class and the membes n the class can be changed dynamcally. Intally, a ecusve data patton s ealzed by membeshp-based locally lnea embeddng data clusteng. Then SVM classfcaton s caed out n each pattoned featue subset. Thus, the authos attempt to solve the classfcaton poblem by fomng multple ease subpoblems. Ths pape pesents a famewok of two ndependent and geneal solutons that am to mpove the pefomance of LDA-based appoaches. Ths methodology s not estcted to face vefcaton, but s able to deal wth any poblem that fts nto the same fomalsm. In the fst step, the dmensonalty of the samples s educed by beakng them down, ceatng subsets of featue vectos wth smalle dmensonalty, and applyng dscmnant analyss on each subset. The esultng dscmnant weght sets ae themselves weghted unde a nomalzaton cteon, thus makng the pecewse dscmnant functons contnuous n ths sense, so as to povde the oveall dscmnant soluton. Ths pocess gves dect mpovements to the two afoementoned poblems as the non-lneaty between the data patten dstbutons s now estcted, wheeas the educed dmensonalty also helps mend the SSS poblem. A sees of smulatons that am to fomulate the face vefcaton poblem llustate the cases fo whch ths method outpefoms tadtonal LDA. Vaous statstcal obsevatons ae made about the dscmnant coeffcents that ae geneated. Remanng stong nonlneates between coespondng subsets lead to a bad estmaton of a numbe of dscmnant coeffcents due to the small tanng set used. These coeffcents ae dentfed and e-estmated n an teatve fashon, f needed. In the second stage, the set of smlaty scoes, that coespond to the efeence mages of each peson, s used n a second dscmnant analyss step. In addton, ths step s complemented by an outle emoval pocess n ode to poduce the fnal vefcaton decson that s a weghted veson of the soted smlaty scoes. The outlne of ths pape s as follows: Secton II descbes the dscmnant poblem at hand n ode to llustate how the poposed famewok contbutes to tacklng a standad face vefcaton poblem. Secton III pesents the two afoementoned stages that compse the novel dscmnant soluton that s poposed n ths pape. Secton IV descbes the stuctue of a sees of smulatons that can be used to povde ndcatons on the expected pefomance of the algothm. Secton V descbes the mplementaton of these smulatons and povdes the coespondng expemental esults. Moeove, n the same secton, the poposed methodology s tested on two well-establshed fontal face databases, the M2VTS and XM2VTS, n ode to assess ts pefomance on standad data sets. The Bussels potocol, whch s used and descbed n [26], was appled to the M2VTS database and Confguaton I of the Lausanne potocol [27] to the XM2VTS database tanng and testng pocedues. II. PROBLEM STATEMENT A wdely known face vefcaton algothm s elastc gaph matchng [28]. The method s based on the analyss of a facal mage egon and ts epesentaton by a set of local descptos (.e. featue vectos extacted at the nodes of a spase gd: j( x = f( x, K, f Μ ( x ( whee f (x denotes the output of a local opeato appled to mage f at the th scale o the th pa (scale, oentaton, x defnes the pxel coodnates and Μ denotes the dmensonalty of the featue vecto. The gd nodes ae ethe evenly dstbuted ove a ectangula mage egon o placed on cetan facal featues (e.g., nose, eyes, etc., called fducal ponts. The basc fom of the mage analyss algothm that was used to collect the featue vectos j fom each face s based on multscale mophologcal dlaton and eoson and s descbed n [26]. All the featue vectos j that have been poduced ae nomalzed n ode to have zeo mean and unt magntude. Let the supescpts and t denote a efeence and a test peson (o gd espectvely. Then, the L nom between the featue vectos at the 2 l th gd node s used as a (sgnal smlaty measue: t C = j( x j( x. (2 l l l

4 TNN05-P485 4 Let c t be a column vecto compsed by the smlaty values between a test and a efeence peson at all L gd nodes,.e.: c Τ = [, (3 t C, K, C ] L In ode to make a decson on whethe a test vecto coesponds to a clent o an mposto, the followng smple dstance measue can be used, whee s an L vecto of ones: D( t, = Τ c. (4 t The fst phase of the algothm poposed n ths pape ntoduces a geneal LDA-based technque that s caed out n the tanng stage and fnds weghts fo each smlaty vecto c t n ode to enhance the dscmnatoy ablty of the dstance measue. As s the case n most face vefcaton applcatons, both the M2VTS and XM2VTS databases, and the potocols they wee evaluated unde, allow fo the fnal decson, of whethe a test facal mage coesponds to a clent o an mposto, to be made by pocessng Τ dffeent mages of the efeence face. That s, the test face s compaed aganst all the mages of the efeence peson contaned n the tanng set. As a esult, we end up wth Τ smlaty, o matchng eo, scoes; tadtonally, the fnal classfcaton decson s based solely on the lowest eo value. The second phase of the poposed algothm povdes an altenatve scoe weghtng method that mpoves the fnal classfcaton ate sgnfcantly. The two methods ae ndependent fom one anothe and ae poposed as geneal solutons fo classfcaton poblems of analogous fom. III. BOOSTING LINEAR DISCRIMINANT ANALYSIS m,c m, I c t Let and denote the sample mean of the class of smlaty vectos that coesponds to clent clams elatng to the efeence peson (nta-class mean and those coespondng to mposto clams elatng to peson (nteclass mean, espectvely. In addton, let Ν and be the C Ν I coespondng numbes of smlaty vectos that belong to these two classes and Ν be the sum,.e., the total numbe of smlaty vectos. Let S and be the wthn-class and W S B between-class scatte matces, espectvely [29]. Suppose that we would lke to tansfom lnealy the smlaty vectos: D ( t, = w Τ. (5 c t The most known and plausble cteon s to fnd a pojecton, o, equvalently, choose w that maxmzes the ato of the between-class scatte aganst the wthn-class scatte (Fshe s cteon: Τ w S Bw J ( w =. (6 Τ w SW w Fo the two-class poblem, as s the case of face vefcaton, Fshe s lnea dscmnant povdes the vecto that maxmzes (6 and s gven by: w, 0 = S W ( m, I m, C. (7 A. Weghted Pecewse Lnea Dscmnant Analyss (WPLDA Model Ou expements, whch ae dscussed n Secton IV, evealed that the tadtonal Fshe s lnea dscmnant pocess not only pefoms pooly, but, n cetan cases, degades the classfcaton capablty of the face vefcaton algothm, patculaly when tanng data fom the M2VTS database was used as can be seen n Table II. That s, the dstance measue n (4 povded a much bette soluton than (5 afte tadtonal LDA was used to detemne the values of w. Ths malady can be attbuted to the nsuffcent numbe Ν C of clent smlaty vectos, wth espect to the dmensonalty L of each vecto ct. Ths s the case fo most face vefcaton poblems and fo the Bussels [26] and Lausanne potocols [27] as well. The fst thng that s done s to povde bette estmaton to Fshe s lnea dscmnant functon. The man poblem s that the class of clent clams s vey small n elaton to the mposto class. Ths fact may affect the tanng [30]. As a esult, a modfed Fshe s lnea dscmnant s used by edefnng (7 to: = Ν I ΝC w, 0 SW m, I m, C Ν Ν, (8 so as to accommodate the po pobabltes of how well the mean of each class s estmated. Secondly, and fo clams elated to each efeence peson, gd nodes that do not possess any dscmnatoy powe ae dscaded. At an aveage 4 nodes, out of 64, ae dscaded fo a 8 8 gd. Smply, each of the L emanng nodes n c t must satsfy: m l l l L. (9, I ( m, C (, =, K, The novelty of ou appoach s that n ode to gve emedy to the SSS poblem, each smlaty vecto c t wth dmensonalty L s boken down to P smalle dmensonalty vectos, c P, each one of length t,, =, K Μ Ν, thus fomng P subsets. The moe Μ, whee ( C c t, statstcally ndependent the vectos ae to each othe, the bette the dscmnant analyss s expected to be. As a esult, P sepaate Fshe lnea dscmnant pocesses ae caed out and each of the weght vectos poduced s nomalzed, so that the wthn goup vaance equals to one, by applyng: Τ 2, 0, = w,0, ( w,0, SW, w,0, w, (0 whee =, K, P s the ndex of Μ dmensonalty vecto c t, coespondng to a subset of smlaty vecto coodnates. Ths nomalzaton step enables the pope megng of all weght vectos to a sngle column weght vecto, w,0, as such: Τ Τ w = w K w. ( [ ] Τ,0,,,0, P, 0,

5 TNN05-P485 5 B. Re-estmatng the Defectve Dscmnant Coeffcents By meetng condton (9, all dscmnant coeffcents that coespond to the emanng gd nodes should ndcate a constuctve contbuton to the oveall dscmnatoy pocess. Snce the matchng eo s always postve and mposto matchng eos should be lage than the clent eos, w should be a vecto of postve weghts only.,0 L The excepton to ths s the possblty to have zeo-valued weghts that would ndcate that cetan gd nodes do not contbute to the classfcaton pocess. In spte of ths, when the set of clent smlaty data pesents ovelap wth the set of mposto smlaty data, such that no sngle lnea hype-plane can sepaate the two classes, t s lkely, dependng on the amount of ovelap and/o how sevee the SSS poblem s, that a numbe of the dscmnant coeffcents n may be w,0 found to be negatve by the dscmnant algothm dung the tanng phase. The WPLDA model that s ntoduced s less susceptble to these occuences, as t settles the SSS poblem. Any negatve dscmnant coeffcents that eman n ae w,0 caused by lage nonlneates of the sepaaton suface between the dstbuton pattens of coespondng subsets and/o the lack of a suffcently lage numbe of tanng samples. By havng the a-poy knowledge that negatve dscmnant coeffcents ae the dect esult of a faulty estmaton pocess and assumng that s the numbe of negatve weghts found n w,0, the followng two cases ae consdeed: Case : L > 0. 5 Μ In ths case, all the gd node tanng data that coespond to the negatve coeffcents n w,0 ae collected and edstbuted nto P Μ dmensonalty vectos whee, deally, each subset agan holds Μ smlaty values. Now, an addtonal dscmnant pocess can be appled on the data contaned n one of the P subsets n ode to poduce Μ new dscmnant weghts. Indeed, P sepaate Fshe lnea dscmnant opeatons ae caed out by usng (8 and each of the P weght vectos of length Μ that s poduced s nomalzed by usng (0. Successvely, all postve weghts fom all P vectos ae collected and used as the fnal multples of c t (dscmnant coeffcents. On the othe hand, all negatve weghts ae collected and once agan tested aganst cases and 2. Ths pocess s caed out n as many teatons as ae equed fo Case 2 to apply. That s, the total numbe of negatve dscmnant weghts should deally be zeo, o at least smalle o equal to Μ. The numbe of teatons n ths pocedue should dop f tanng s caed out on the low ( Μ dmensonalty smlaty vectos, as opposed to havng been appled on the full L dmensonal vectos. Case 2: 0. 5 Μ L L All negatve weghts ae set equal to zeo and no futhe pocessng s equed. The facto 0.5 s used to ndcate that, f the numbe of smlaty values, n the fnal Μ dmensonalty vecto that holds smlaty values coespondng to negatve dscmnant coeffcents, s not equal to moe than half of ts full capacty Μ, the coespondng lnea dscmnant equaton depends on too few vaables and s lkely to gve lage naccuaces to the oveall dscmnant soluton. In ode to avod buldng the oveall dscmnant soluton by also ncludng these lage naccuaces, whch wll essentally tanslate to defectve dscmnant coeffcent values, zeo weghts ae assgned to ndcate that these coeffcents no longe have any dscmnant sgnfcance. C. Weghtng the Multple Classfcaton Scoes Most, f not all, face vefcaton applcatons allow fo a test peson to be classfed as an mposto o a clent by usng numeous mages of the efeence face. Thus, numeous vefcaton tests ae caed out wheneve a clam s consdeed. As a esult, multple (Τ classfcaton scoes D ( t, d, whee d =, K, T, ae avalable fo each clam of an dentty, by a test peson t. Tadtonally, the test peson t s classfed as a clent f the mnmum value out of the total scoes, D ( t, = mn D ( t,, d =, K, T, s below a Τ { } mn =, KT pedefned theshold, and as an mposto, f t s above ths theshold. In ths wok, tanng data ae used once agan to deve peson specfc weghts to be used fo the combnaton of the Τ scoes. The motvaton behnd ths pocess s that, deally, all Τ scoes should contbute to the fnal classfcaton decson, as, n cetan cases, the mposto mage that coesponds to a mnmum scoe may have accdentally - e.g., due to a patcula facal expesson o due to smla eyeglasses- had close smlaty to a cetan efeence mage. In such a case, the emanng efeence mages can be used n an effot to epa the false classfcaton decson. Now the poblem becomes: Τ d = d D ( t, = v D ( t, (2, d d Agan, Fshe s modfed lnea dscmnant (8 s appled to detemne the vecto v whch contans the Τ weghts, d =, T, of the classfcaton scoes. v, d K, A much lage numbe of mposto, athe than clent, smlaty scoes s usually avalable n the tanng set of a face vefcaton database. Ths nceases the pobablty that some mposto mages may andomly gve a close match to a efeence photo, even close than some of the clent mages gve. Wheneve ths happens, the pocess of estmatng a sepaaton between the two classes degades sgnfcantly because of the small numbe of clent tanng smlaty scoes, whch equals to the numbe of tanng samples n secton III-A. Thus, an outle emoval pocess s ncopoated, whee the mnmum mposto smlaty scoes n the tanng set of each efeence peson,.e. all D t, (

6 TNN05-P485 6 scoes that coespond to mposto matches, ae odeed and the smallest Q% of these values ae dscaded. As a esult, the lnea dscmnant pocess gves a moe accuate sepaaton that helps ncease the classfcaton pefomance. IV. SIMULATED AND EXPERIMENTAL RESULTS In ths secton, the effcency of the poposed dscmnant soluton s evaluated usng both smulated and eal data sets. The smulated data sets ae used n ode to deduce expemental evdence on the pefomance of WPLDA, wheeas the eal data that ae taken fom the M2VTS and XM2VTS databases ae used to test the classfcaton ablty of the oveall dscmnant algothm that s pesented n ths pape. A. Classfcaton Pefomance on Smulated Data In ode to povde elevant backgound on the expected pefomance of the poposed WPLDA algothm n face vefcaton, smulatons that tackle the 2-class poblem ae caed out. We ntent to nvestgate the cases whee one can expect the WPLDA algothm to outpefom the tadtonal LDA algothm, wth espect to the sze of the mposto and clent classes. Fo each vefcaton expement, two classes of matchng vectos, one that coesponds to the clents and the othe to the mpostos, ae ceated. Each class contans N sample vectos of dmensonalty L. Each of these sample vectos contans entes dawn fom a nomal (Gaussan dstbuton. The L andom entes to each sample vecto of class Χ, whch s the th clent o mposto class, ae j j geneated by j j 2 ( x μ j 2 j j j 2 ( x e ( σ Ν L j : μ, σ =, = K, (3 Χ j σ 2π j j whee μ G + α and σ K + β. G s the = j = j j K j expected mean value and the expected standad devaton fo the th andom enty of the j th class and s a andom numbe, chosen fom a nomal dstbuton wth zeo mean and unt vaance. The scalas α and β affect the unfomty among the vectos of each class. The dmensonalty of the sample vectos s set to L = 64 n ode to be dentcal to the dmensonalty of the featue vectos - o to the numbe of gd nodes - of the eal face vefcaton poblem that we ae tyng to solve n secton IV- C. Each class contans N = 2000 sample vectos. Let I be the mposto class and C and C2 be two clent classes. Let the andom entes to each sample vecto of the mposto class I and the clent classes C and C2 be geneated based on the followng nomal dstbutons, espectvely: Ν I ( x : μ = , σ = , = K64. (4 Ν ( : = , = , = K64. (5 C x μ σ Ν ( x : μ = , = , = K64. (6 C2 σ It s clea that the mean of the andom entes of s expected to devate moe, w..t. the mean of the entes of C, fom the mean of the entes of I. When the elastc gaph matchng algothm s appled to face vefcaton tasks, t s expected that cetan nodes should povde moe dscmnant nfomaton than othes. Ths s also tue fo most featue-based vefcaton methods. Fo example, n geneal a node that les at the locaton of the nose wll be moe useful than a node that les at a locaton on the foehead. In ode to smulate a smla stuaton, we ceate a subset of LB nodes (out of the total L, that s expected to be moe dscmnant than the emanng nodes. We name ths set of L B nodes as most dscmnant coeffcents. Let a clent class C be ceated, such that the entes at the nodes ae 3 LB taken fom the C clent class (snce the entes fom ae 2 C2 moe sepaated fom the entes n I than the entes of C ae and the est of the node entes fom the C class. Fo ths fst set of expements we let L B = 5 and the postons of the 5 most dscmnant coeffcents ae selected so as to be evenly spaced fom one anothe, e.g. the coeffcent ndex s gven by {,22,33,44,55} fo L = 64. The data that wee ceated ae used to compae the dscmnaton ablty of tadtonal LDA and the poposed WPLDA fo vaous numbes of tanng sample vectos fo the mposto and clent class. Fo each 2-class poblem that s fomulated, one tanng and one test set ae ceated. The tanng set of LDA and WPLDA s fomed based on the andom selecton out of the complete set of N sample vectos of each class. The emanng sample vectos of each class, obtaned by excludng the tanng set of LDA and WPLDA, fom the test set that s used to evaluate the classfcaton pefomance. In ode to appoxmate the deal lnea dscmnant soluton, a thd method that wll be efeed to as Ideal LDA (ILDA wll always apply the tadtonal LDA algothm makng use of the complete sets of N clent sample vectos and N mposto sample vectos, dung the tanng phase. We consde ths numbe of samples to be lage enough fo the tadtonal LDA algothm to poduce a statstcally coect dscmnant soluton. The test set whee the pefomance of ILDA wll be evaluated on, s dentcal to the test set of LDA and WPLDA. Thus, the test set s always ncluded n the tanng set of ILDA, so as to best appoxmate the deal lnea dscmnant soluton and povde gound-tuth esults. In addton, and agan fo compason puposes, the classfcaton pefomance of a fouth method wll be consdeed, whee ths method smply computes the mean of the sample vectos (MSV and poduces a non-weghted esult whch can be used to ndcate how dffcult the 2-class classfcaton poblem s. In ode to evaluate the pefomance of the fou afoementoned methods the equal eo ate (EER s employed. Each of the EER values epoted has been aveaged C 2

7 TNN05-P485 7 ove 20 ndependent uns of an dentcal expement fo moe accuate esults. The smulaton data ae used n vaous dscmnant pocesses that am to sepaate out the clent and mposto classes. The 2-class poblem that s studed next uses data fom I and C 3. Fg. -3 show the EER when the numbe of clent sample vectos vaes fom 2 to 00. It s noted that logathmc scales ae used fo the y-axs. Fg. shows the EER esults when the numbe of mposto sample vectos s 0. Fo the LDA algothm, the SSS s expected to have the most sevee effects on the EER when the clent class has less than ( L + = 65 samples. In theoy, n ths case nethe the clent class no the mposto class can be popely modelled by tadtonal LDA and, as a esult, an appopate sepaaton between the two classes cannot be found. On the othe hand, WPLDA s not affected by the SSS poblem as can be seen n Fg.. The small vaatons n the EER of ILDA ndcate the amount of andomness n ou esults snce only the y-axs showng EER s sgnfcant fo the ILDA esults. Fg. 2 and 3 show the EER ates fo 00 and 000 mposto sample vectos espectvely. It s clealy seen n these fgues that, unless a elatvely lage numbe of clent and mposto sample vectos ae avalable, WPLDA outpefoms LDA. Fg. 2 shows that, when 00 mposto and 83 clent sample vectos ae avalable, the pefomance of LDA becomes bette than that of MSV. Fg. 3 shows that when the numbe of mposto sample vectos becomes 000, 20 clent sample vectos ae equed fo LDA to outpefom WPLDA. Fo most cuent bometc databases, havng 20, o moe, clent samples pe peson s qute uncommon. Fg. 3 also shows that when the clent and mposto class szes ae lage enough such that tadtonal LDA can fnd a pope estmaton of a lnea sepaaton hype-plane between the classes, tadtonal LDA pesents a stonge classfcaton pefomance snce the pope hghe-dmensonalty soluton s moe geneal than the lowedmensonalty solutons offeed by WPLDA. Fo efeence, t s stated that n smulatons we un whee the clent class conssted only sample vectos fom ethe C o the C2 aveage dop n the EER ate of LDA when 000 mposto sample vectos ae used nstead of 0 s 25.50%, wheeas fo WPLDA 0.37%. Of couse, t s expected that a qute lage numbe of mposto sample vectos ae equed fo the LDA algothm to outpefom WPLDA when the numbe of clent sample vectos s only 6, as s n the face vefcaton poblem that we ae tyng to solve n secton IV-C. As a esult, we expect that WPLDA should povde bette vefcaton pefomance. B. Dscmnant Chaactestcs unde the SSS Poblem The second set of expements usng smulated data nvolves nvestgatng the statstcal behavou of the dscmnant coeffcents of the LDA and WPLDA pocesses wth efeence to ILDA. Moeove, EER ates ae epoted fo dffeent numbes of most dscmnant coeffcents contaned n each class, that s, fo vaous values of L B. The L B most dscmnant coeffcents ae evenly spead out, as much as possble, n the L -dmensonal space. In ode to detemne how effcent each dscmnant method s n ecognzng the mpotance of the most dscmnant coeffcents, a sepaaton cteon between the most dscmnant and the emanng coeffcents s defned as: mb mr H =, (7 sb + s R whee mb and sb ae scalas epesentng the aveage mean and the aveage standad devaton of the set of most dscmnant coeffcents and mr and s R those of the emanng coeffcents. If H, the sepaaton cteon s satsfed, snce then the values of the most dscmnant coeffcents vay sgnfcantly fom those of the emanng coeffcents. Based on the pactcal consdeatons of the face vefcaton poblem at hand, ths set of smulatons s modelled unde the SSS poblem, whee the clent class has less sample vectos than the dmensonalty of the smlaty vectos. The M2VTS and XM2VTS face vefcaton test potocols specfy fo the clent class to aval 6 tanng samples and fo the mposto class to aval 20 o 79 tanng samples fo the M2VTS and XM2VTS databases espectvely. Theefoe, n ode to coelate the smulaton esults wth the expected pefomance of the MSV, LDA and WPLDA algothms n these potocols, we andomly select 6 sample vectos fom the C 3 clent class and 000 sample vectos fom the I mposto class to tan LDA and WPLDA. The coeffcents of ILDA ae once agan geneated by a tanng set of 2000 clent and 2000 mposto sample vectos. To obseve the statstcal behavou of the dscmnant coeffcents, 000 ndependent uns wee caed out. The entes at the poston of the L B elements ae expected to have a lage dstance fom the coespondng element entes of class I, than the est. As a esult, the dscmnant pocess should gve lage weghts fo the element entes at these L B specfc postons, snce they ae expected to be the most useful n poducng a meanngful sepaaton between the mposto and the clent class. Fg. 4,5 and 6 show the boxplots [3] that povde statstcal nfomaton about the calculaton of the 64 dscmnant coeffcents, w, = K64, thoughout the 000 ndependent uns, by ILDA, LDA and WPLDA espectvely. These thee methods pocessed the C, (wth, and 3 LB = 5 I tanng data. The boxes have lnes at the lowe quatle, medan, and uppe quatle values. The whskes ae lnes extendng fom each end of the boxes to show the extent of the est of the data, specfed as.5 tmes the nte-quatle ange. Outles ae data wth values beyond the ends of the whskes and ae ndcated usng +. It s clea that WPLDA, unlke LDA, povdes a complete sepaaton to all the most dscmnant coeffcents fom the emanng coeffcents, n tems of assgnng lagest weghts, wheeas ILDA almost does the same. Results such as the ones shown n Fg. 4 though 6

8 TNN05-P485 8 wee poduced fo vaous values of and the coespondng EER and H values (7 wee calculated. These esults ae summazed n Table I. Once agan, the case whee half the coeffcents ae the most dscmnant povdes an excepton to ou esults snce, n all othe cases, WPLDA s elated wth the lagest H value. In addton, WPLDA povdes the EER ate that s closest to the coespondng ILDA ate. The EER of WPLDA and MSV s dentcal n the cases whee no most dscmnant coeffcents exst,.e. when as many as half the coeffcents ae most dscmnant ones. As expected, WPLDA always shows a bette classfcaton pefomance than tadtonal LDA, unde the SSS poblem. C. Pefomance Evaluaton on the M2VTS and XM2VTS Databases. In ths secton, expemental tests ae caed out by applyng the testng potocols of the M2VTS and XM2VTS databases. The M2VTS database contans vdeo data of 37 pesons. Fou ecodngs/shots of the 37 pesons have been collected, each contanng a fontal-face pose. The Bussels potocol, whch s used n [26, 29] eques the mplementaton of fou expemental sessons by employng the leave-one-out and otaton estmates. In each sesson, one shot s left out to be used as the test set. In ode to mplement test mposto clams, otatons ove the 37 peson denttes ae caed out by consdeng the fontal face mage of each peson n the test set as mposto. By excludng any fontal face mage of the test mposto fom the emanng thee shots, a tanng set that conssted of 36 clents s bult. The test mposto petends to be one of the 36 clents and ths attempt s epeated fo all clent denttes. As a esult, 36 mposto clams ae poduced. In a smla manne, 36 test clent clams ae tested by employng the clent fontal faces fom the shot that s left out, and those of the tanng set. The tanng pocedue s analogous to the test pocedue that was just descbed. It s appled to the tanng set of the 36 clents. Thee fontal face mages ae avalable fo each clent. By consdeng all pemutatons of the thee fontal mages of the same peson, taken two at a tme, 6 tanng clent clams can be mplemented. Moeove, 20 tanng mposto clams, when each of the othe 35 pesons attempt to access the system wth the dentty of the peson unde consdeaton, ae mplemented. That s, anothe 6 aw smlaty vectos coespondng to all pa-wse compasons between the fontal mages of any two dffeent pesons taken fom dffeent shots ae poduced. Fo a moe detaled descpton of the Bussels potocol the eade s efeed to [26]. The XM2VTS database contans fou ecodngs of 295 subjects taken ove a peod of fou months. The Lausanne potocol descbed n [27] splts andomly all subjects nto clent and mposto goups. The clent goup contans 200 subjects, the mposto goup s dvded nto 25 evaluaton mpostos and 70 test mpostos. Eght mages fom 4 sessons ae used. Fom these sets consstng of face mages, a tanng set, an evaluaton set and a test set ae bult. Thee exst two L B confguatons that dffe n the selecton of patcula shots of people nto the tanng, evaluaton and test sets. The tanng set of the Confguaton I contans 200 pesons wth 3 mages pe peson. The evaluaton set contans 3 mages pe clent fo genune clams and 25 evaluaton mpostos wth 8 mages pe mposto. The test set has 2 mages pe clent and 70 mpostos wth 8 mages pe mposto. The tanng set s used to constuct clent models. The evaluaton set s selected to poduce clent and mposto access scoes, whch ae used to fnd a theshold that detemnes f a peson s accepted o not as a clent (t can be a clent-specfc theshold o global theshold. Accodng to the Lausanne potocol the theshold s set to satsfy cetan pefomance levels (eo ates on the evaluaton set. Fnally the test set s selected to smulate ealstc authentcaton tests whee the mposto dentty s unknown to the system [32]. Fo a moe detaled descpton of the Lausanne potocol the eade s efeed to [27]. The poposed methodology s now evaluated usng the Bussels and Lausanne standad potocols descbed above that, as s the case wth most face vefcaton applcatons, suffe fom the SSS poblem. Specfcally, the numbe of clent smlaty vectos Ν C fo each ndvdual that wee avalable n the tanng set was only 6, wheeas the 8 8 gd that was used set the dmensonalty of the smlaty vecto to L = 64. The value of Ν I was set to 20 and 79, when tanng the algothm usng M2VTS and XM2VTS data espectvely. Fo the XM2VTS database and Confguaton I of the Lausanne potocol, a total of 600 (3 clent shots x 200 clents clent clam tests and 40,000 (25 mpostos x 8 shots x 200 clents mposto clam tests wee caed out fo the evaluaton set and 400 (2 clent shots x 200 clents clent clams and 2,000 (70 mpostos x 8 shots x 200 clents mposto clams fo the test set. Fo the M2VTS database and the Bussels potocol, a total of 5,328 clent clam tests and 5,328 mposto clam tests ( clent o mposto x 36 otatons x 4 shots x 37 ndvduals wee caed out. Face vefcaton decson thesholds fom the coespondng tanng pocess of each database wee collected and used to evaluate the vefcaton esults, except fo the evaluaton of the XM2VTS test set, whee thesholds fom the evaluaton pocess wee used, as [27] suggests. The dscmnant coeffcent vectos w deved by the pocesses descbed n sub-sectons III-A and III-B have been used to wegh the nomalzed smlaty vectos c that ae povded by the Mophologcal Elastc Gaph Matchng pocedue appled to fontal face vefcaton, based on the algothm descbed n [26]. Ou tests evealed that the optmum value fo Μ s 4. Moeove, t was obseved that, on the aveage, 36.54% of the dscmnant coeffcents n and 6.27% of the dscmnant coeffcents n wee w,0 found to be negatve fo the M2VTS tanng set. Addtonally, 24.39% of the dscmnant coeffcents n and 0.76% of the dscmnant coeffcents n w,0 w,0 w,0 wee

9 TNN05-P485 9 found to be negatve, when the lage XM2VTS tanng set was used. In addton, dung the tanng stages of the M2VTS database, 3 to 5 teatons descbed n sub-secton III-B, ae usually equed when Μ = L, wheeas no moe than 2 teatons ae equed when Μ s set to 4. Fo the latte value of Μ, one, at the most, teaton s needed when pocessng XM2VTS data. The pocedue descbed n Secton III-C s used to calculate a moe accuate smlaty scoe fo each tested ndvdual. The testng potocols specfy that a test peson can be classfed to be an mposto o a clent by usng thee dffeent mages of the efeence peson ( Τ = 3. Thus, thee tests ae caed out and thee smlaty scoes ae avalable fo each ndvdual. Unfotunately, the tanng data whch we can wok wth to deve these weghts only povde two combnatons, snce a total of 6 tanng clent combnatons ae avalable fo the 3 dffeent mages of each peson. Thus, we ae foced to set the lagest smlaty scoe to zeo and set Τ = 2 n (2. The two weghts ae found usng (8. Fo the outle emoval pocess, Q s set to 4, that s, 4% of the mnmum mposto smlaty scoes s dscaded. Let us denote the combnaton of the mophologcal elastc gaph matchng, (MEGM, and the weghtng appoach that makes up fo the fst phase of the poposed algothm, as s descbed n sub-sectons III-A and III-B, by, once agan, WPLDA. Moeove, let MS-WPLDA be the second phase of the algothm that s appled on WPLDA and s descbed n sub-secton III-C, whee MS stands fo multple scoe. In ode to evaluate the pefomance of these methods the FAR and FRR ate measues ae used. Fg. 7 shows a ctcal egon of the ROC cuves fo the aw MEGM data usng (4, classcal LDA (7 appled on the aw MEGM data, WPLDA and MS- WPLDA evaluated on the M2VTS database. Fg. 8 shows the same coespondng ROC cuves when the algothms wee evaluated on the XM2VTS evaluaton set and Fg. 9 the coespondng ones fo the XM2VTS test set. Results ae pesented n logathmc scales. In addton, Table II shows the EER fo each algothm. When the M2VTS data ae used, the tadtonal LDA algothm degades the classfcaton pefomance sgnfcantly, havng a poo genealzaton ablty, whch stems fom the lagely nadequate, n tems of sze, tanng set that was avalable. Tadtonal LDA undepefoms, wth espect to WPLDA, at a lage degee on the M2VTS, athe than on the XM2VTS, expements. Ths can be attbuted to the lage data set that s used n the XM2VTS tanng pocess. In addton, the expemental esults show that the poposed WPLDA algothm pefoms bette than ethe MEGM o LDA, as was pevously ndcated by the smulaton esults of secton IV. Futhemoe, the ndependent MS-WPLDA pocess povdes addtonal mpovement to the classfcaton ablty of WPLDA. In ode to compae the pefomance of WPLDA wth a state-of-the-at method t s mpotant to select an algothm whch s expected to pefom well not only unde the SSS poblem, but also when dealng wth the 2-class poblem. Fo example, the algothms n [20] and [23] ae desgned unde the assumpton of a mult-class poblem. On the contay, the RLDA algothm that was ecently poposed n [8] s not desgned aound the mult-class poblem. Thus, we apply the Bussels and Lausanne face vefcaton potocols to evaluate ts pefomance. Fo salent compasons, we epot esults geneated by the RLDA algothm afte dscadng the nodes that do not possess any dscmnatoy powe, by makng use of (9. The EER pefomance of RLDA s shown n Table II. These esults llustate that WPLDA always gves bette classfcaton esults that the RLDA algothm. It s antcpated that the bas ntoduced by the egulazaton n ode to educe the hgh vaance elated to the egenvalue estmates of the wthn-class scatte matx lmts the classfcaton accuacy of RLDA (essentally due to havng nsuffcent samples to epesent the clent class, wheeas WPLDA acheves a bette soluton snce t decomposes the dscmnant analyss poblem nto multple lowe-dmensonalty poblems. V. 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Messe et. al., Face vefcaton competton on the XM2VTS database, n Poc. 4 th Int. Conf. Audo-and-Vdeo-Based Bometc Peson Authentcaton, Guldfod, UK, June 9-, Maos Kypeountas eceved the B.Sc. n electcal engneeng n 2002 and the M.Sc. n electcal engneeng n 2003, both fom Floda Atlantc Unvesty n Boca Raton, Floda. He was a eseach assstant at the FAU Imagng Technology Cente fom 2000 untl 2003 whee he woked on seveal hgh-esoluton magng R&D pojects funded by NASA, DARPA and the US NAVY. Cuently, he s a PhD student at the Atfcal Intellgence and Infomaton Analyss lab of the Depatment of Infomatcs at the Astotle Unvesty of Thessalonk. Hs eseach nteests nclude hgh esoluton and ultasonc magng, patten ecognton, DSP algothms and eal-tme vdeo pocessng. Anastasos Tefas (M 04 eceved the B.Sc. n nfomatcs n 997 and the Ph.D. degee n nfomatcs n 2002, both fom the Astotle Unvesty of Thessalonk, Geece. Snce 2006, he has been an Assstant Pofesso at the Depatment of Infomaton Management, Technologcal Educatonal Insttute of Kavala. Fom 997 to 2002, he was a eseache and teachng assstant n the Depatment of Infomatcs, Unvesty of Thessalonk. Fom 2003 to 2004, he was a tempoay lectue n the Depatment of Infomatcs, Unvesty of Thessalonk whee he s cuently, a seno eseache. He has co-authoed ove 50 jounal and confeence papes. Hs cuent eseach nteests nclude computatonal ntellgence, patten ecognton, dgtal sgnal and mage pocessng, detecton and estmaton theoy, and compute vson. Ioanns Ptas (SM 94 eceved the Dploma of Electcal Engneeng n 980 and the PhD degee n electcal engneeng n 985, both fom the Unvesty of Thessalonk, Geece. Snce 994 he has been a Pofesso at the Depatment of Infomatcs, Unvesty of Thessalonk, Geece. Fom 980 to 993 he seved as Scentfc Assstant, Lectue, Assstant Pofesso, and Assocate Pofesso n the Depatment of Electcal and Compute Engneeng at the same Unvesty. He seved as Vstng Pofesso and ASI fellow at the Unvesty of Btsh Columba, Canada, as Vstng Pofesso at Ecole Polytechnque Fedeale de Lausanne, at Tampee Unvesty of Technology, Fnland, as Vstng Assstant Pofesso at the Unvesty of Toonto and as a Vstng Reseach Assocate at the Unvesty of Toonto, Canada and at the Unvesty of Elangen-Nuenbeg, Gemany. He has publshed ove 50 papes, contbuted n 20 books and authoed, co-authoed, edted, co-edted 7 books n hs aea of nteest. He s the coautho of the books Nonlnea Dgtal Fltes: Pncples and Applcatons (Boston, MA: Kluwe, 990, and 3-D Image Pocessng Algothms(New Yok: Wley,2000, Nonlnea Model- Based Image/Vdeo Pocessng and Analyss (New Yok: Wley, 200; and autho of Dgtal Image Pocessng Algothms and Applcatons (New Yok: Wley, He s the edto of the book Paallel Algothms and Achtectues fo Dgtal Image Pocessng, Compute Vson and Neual