Discriminative Hessian Eigenmaps for face recognition

Transcription

1 tle Dscrmnatve essan Egenmaps or ace recognton Author(s) S S; ao D; Chan KP Ctaton he 200 IEEE Internatonal Conerence on Acoustcs Speech and Sgnal Processng (ICASSP) Dallas. 4-9 March 200. In IEEE Internatonal Conerence on Acoustcs Speech and Sgnal Processng Proceedngs 200 p Issued Date 200 UR Rghts IEEE Internatonal Conerence on Acoustcs Speech and Sgnal Processng Proceedngs. Copyrght IEEE.; 200 IEEE. Personal use o ths materal s permtted. owever permsson to reprnt/republsh ths materal or advertsng or promotonal purposes or or creatng new collectve works or resale or redstrbuton to servers or lsts or to reuse any copyrghted component o ths work n other works must be obtaned rom the IEEE.; hs work s lcensed under a Creatve Commons Attrbuton-NonCommercal-NoDervatves 4.0 Internatonal cense.

2 DISCRIMINAIVE ESSIAN EIGENMAPS FOR FACE RECOGNIION S S Dacheng ao 2 Kwok-Png Chan Department o Computer Scence Unversty o ong Kong ong Kong 2 School o Computer Engneerng Nanyang echnologcal Unversty Sngapore ABSRAC Dmenson reducton algorthms have attracted a lot o attentons n ace recognton because they can select a subset o eectve and ecent dscrmnatve eatures n the ace mages. Most o dmenson reducton algorthms can not well model both the ntra-class geometry and nterclass dscrmnaton smultaneously. In ths paper we ntroduce the Dscrmnatve essan Egenmaps () a novel dmenson reducton algorthm to address ths problem. wll consder encodng the geometrc and dscrmnatve normaton n a local patch by mproved essan Egenmaps and margn mamzaton respectvely. Emprcal studes on publc ace database thoroughly demonstrate that s superor to popular algorthms or dmenson reducton e.g. FDA PP and DA. Inde erms Dmenson Reducton Manold earnng Face Recognton.. INRODUCION Dmenson reducton [3][] plays an mportant role n varous tasks n computer vson e.g. ace recognton. A key role or ace recognton s the dstance or smlarty between ace mages whch can be solved va dmenson reducton as dmenson reducton perorms the recognton by enlargng the smlarty among the ntra-class samples and mamzng the derence among the nter-class samples n a subspace rather than the orgnal eature space. A dmenson reducton algorthm projects the orgnal hgh-dmensonal eature space to a low-dmensonal subspace where specc statstcal propertes can be well preserved. For eample prncple component analyss (PCA) [] one o the most popular unsupervsed dmenson reducton algorthms mamzes the varance o the data n the projected subspace; Fsher s lnear dscrmnatve analyss (FDA) [2] the most tradtonal supervsed dmenson reducton algorthm mnmzes the trace rato between the wthn class scatter and the between class scatter so that the Gaussan dstrbuted samples can be well separated n the selected subspace; localty preservng projectons (PP) [4] preserves the local geometry o samples by processng an undrected weghted graph that represents the neghbourhood relatons o parwse samples; Margnal Fsher analyss () [2] consders both the ntra-class geometry and nteracton o samples rom derent classes; Dscrmnatve localty algnment (DA) [5] preserves the dscrmnatve normaton by mamzng the dstance among the nter-class samples and mnmzng the dstance among the ntra-class samples over the local patch o each sample. owever the geometrc and dscrmnatve normaton n these dmenson reducton algorthms are not well modeled e.g. DA does not consder the geometrc normaton; gnores the dscrmnatve normaton o non-margnal samples rom derent classes. By usng the patch algnment ramework [6] we can model both the ntra-class local geometry and the nter-class dscrmnatve normaton convenently. In partcular or each sample and ts assocated patch (neghbours o the sample) t s mportant to consder the ollowng two propertes: ) the ntra-class local geometry can be represented by the local tangent space whch s locally sometrc to the manold o the ntra-class nearest samples o the patch; and 2) the nter-class dscrmnatve normaton can be represented by the margn between the ntra-class neghbor samples and the nter-class nearest samples o the patch. Because the method used or local geometry representaton s smlar to essan Egenmaps [7] the proposed dmenson reducton algorthm s termed the Dscrmnatve essan Egenmaps or or short. he rest o ths paper s organzed as ollows. Secton 2 ntroduces the proposed Dscrmnatve essan Egenmaps (). Secton 3 shows the results o thoroughly emprcal studes. Secton 4 concludes. 2. DISCRIMINAIVE ESSIAN EIGENMAPS hs Secton presents the dscrmnatve essan Egenmaps or or short to solve the ace recognton tasks. In D d we try to nd an optmal lnear mappng W R D so that t can project R to a low-dmensonal space as d y W R. In ths learned low-dmensonal space characterzes two specc propertes: /0/$ IEEE 5586 ICASSP 200

3 . he local geometry property - nearby samples n the orgnal Eucldean space are close to each other n the learned subspace. 2. he dscrmnatve property - samples rom derent classes can be well separated n the learned subspace. In summary the dscrmnatve normaton as well as the local geometry wll be well modeled n the. 2.. Moded essan Egenmaps Emprcally ntra-class geometry s useul or classcaton. essan Egenmaps [7] s a geometry preservaton manold learnng method that can recover the underlyng parameterzaton o a manold M embedded n a hghdmensonal space the manold M s locally sometrc to d an open and connected subset o R. Because the parameter space need not be conve n essan Egenmaps t can be appled to model a nonconve manold e.g. an S- curve surace wth a hole. hereore we adapt essan Egenmaps n to preserve the local geometry or dmenson reducton. essan Egnmaps nds the (d+)-dmensonal nullspace o where s the essan matr o a smooth mappng.e. : M R can be. hs calculated by usng 2 d wheren s the essan o on the patch k and the correspondng output n low-dmensonal space s Y y y y k. he M a Eucldean space tangental to M at tangent plane s an orthogonal coordnate system. In order to estmate we calculate the local coordnate system o and each sample n on the tangent plane M has ts own local coordnate can be estmated by usng. owever essan Egenmaps cannot be appled to many practcal applcatons e.g. ace recognton because t requres that k d where k s the number o the neghbourng samples and d s the dmenson o the subspace. It s dcult to guarantee ths condton because M. Aterwards ths we have a lmted number o samples. We propose to overcome ths problem by perormng PCA on M at and orthnormalzng the d-dmensonal representaton to obtan the tangent coordnate n F M. he ollowng steps or the moded essan Egenmaps are smlar to those n essan Egenmaps. Under the patch algnment ramework the objectve uncton or the moded essan Egenmaps to preserve the local geometry on a local patch Y can be wrtten as where normaton o the patch and y tr Y Y tr Y Y () encodes the local geometry y s the local geometry representaton. Under the help o local geometrc normaton can be urther preserved Margn Mamzaton As or classcaton however t s nsucent to only retan the local geometry because no labelng normaton s taken nto account. o urther eplot the dscrmnatve power lke the denton o the local geometry we can dene a new margn mamzaton [3] based scheme or dscrmnatve normaton preservaton over patches. In partcular or each sample assocated wth a patch k M k wheren 2 k.e. the k nearest samples o are rom the same class as and.e. the other k 2 nearest samples o are rom derent classes aganst we dene the margn as the average derence between two knds o dstances on ths patch. One s called nter-class dstance that s the dstance between and samples takng derent labels.e. ; the other s called ntra-class dstance that s the dstance between and samples sharng the same label.e. k. Bascally n the patch M s lowdmensonal representaton Y y y y k y M y k 2 we epect the margn between ntra-class and nter-class samples wll be mamzed as large as possble.e. 2 k 2 y y y j p y p k. (2) 2 j k On the other hand based on (2) we try to mnmze the bellowng objectve uncton: k 2 2 M y y y j y yp j k p e k tr Y dag k M w e k I 2 kk Y 2 M I (3) k k 2 tr Y Y where M M M k w / k.../ k -/k /k 2 ; Ik s the 5587

4 2 k k k k dentty matr ;... k k e R 2 2 k M normaton representaton. ; k w w j j and w dag w M y s the margn 2.3. Dscrmnatve essan Egenmaps () By usng the results obtaned rom the prevous subsectons we can obtan the optmzaton ramework to learn the projecton matr W whch can utlze both the local geometry and the dscrmnatve normaton. Because the margn representaton M y and the local geometry representaton y are dened over patches and each patch has ts own coordnate system algnment strategy s adopted here to buld a global coordnate or all patches dened or the tranng samples. As a consequence the objectve uncton or to solve the dmenson reducton problem s gven by l W arg mn M y y (4) Dd WR where s the tunng parameter. I we dene two selecton matres S and S M whch select samples n the th patch rom all the tranng samples Y y y2 yl or constructng M y and y respectvely. hereore Y Y S and Y Y M SM wth Y representng the patch or the local geometry preservaton and Y M denotng the patch or margn mamzaton. Ater pluggng () and (3) the objectve uncton n (4) wll turn to l tr Y Y M M M W arg mn 5 Dd WR tr Y Y tr YS M l M YS M arg mn Dd WR tr YS YS l SM MS M arg mn tr Y Y Dd WR S S arg mn tr YY Dd WR l where S S M M MS S s the algnment matr encodng both the local geometry and the dscrmnatve normaton. For lnearzaton Y W s usually consdered where W s the projecton matr. We can mpose derent constrants e.g. Y Y I or W W I to unquely determne Y. he constrant W W I wll be adopted throughout the paper. Under ths constrant and Y W the soluton o (5) can be obtaned by usng the conventonal agrangan multpler method [0] or the generalzed egenvalue decomposton [8]. 3. EPERIMENS In ths Secton we justy our proposed algorthm wth our representatve dmenson reducton algorthms whch are the Fsher s lnear dscrmnant analyss (FDA) [2] the localty preservaton projectons (PP) [4] wth the supervsed settng the margnal Fsher s analyss () [2] and dscrmnatve localty algnment (DA) [6] or ace recognton based on a publc database: CMU-PIE dataset [9]. Fgure. Sample mages rom CMU-PIE database he CMU-PIE dataset contans 4368 mages o 68 people under 3 derent poses 43 derent llumnaton condtons and 4 derent epressons and we randomly select 0 mages per ndvdual n the CMU-PIE dataset n ths eperment. Eample ace mages rom the CMU-PIE database are shown n Fgure. he mages rom CMU-PIE used or our eperments are o sze 3232 n raw pel. In the tranng stage we learn the projecton matr W rom each nvolved algorthm on the tranng samples. In the testng stage each testng sample wll be projected nto the low-dmensonal space by W and ater that nearest-neghbor rule (NN) s appled to predct label o the test mage n the selected subspace. We randomly select p (= 4 5 6) mages per ndvdual or tranng n the database and use the remanng mages or testng. All trals are repeated ten tmes and then the average recognton rates are calculated. Fgure 2 shows the results o aganst FDA PP and DA wth regard to ace recognton accuracy under derent dmensons. able provdes the best recognton rate or each algorthm. It also provdes the optmal values o k k 2 and or whch are tuned by the cross valdaton. 5588

5 00 4 ran 00 5 ran 00 6 ran Recognton Rate (%) FDA PP DA Subspace dmensons Recognton Rate (%) FDA PP DA Subspace dmensons FDA PP DA Subspace dmensons Fgure 2. Recognton rate vs. dmenson reducton on the CMU-PIE database under derent splts. Recognton Rate (%) able. Best recognton rates (%) on CMU-PIE database. he numbers n the parentheses are the subspace dmensons. For he numbers n the parentheses rom let to rght are the subspace dmensons k and respectvely. FDA PP DA 4 ran 8.79(67) 82.33(68) 88.58(78) 86.(39).86(2936) 5 ran 88.94(67) 89.38(67).29().94(62) 94.44(47365) 6 ran 92.58(69) 93.7(67) 92.67() 93.46(34) 96.8(473) As shown n Fgure 2 and able outperorms conventonal algorthms or at least can obtan a comparable perormance n comparng wth the conventonal algorthms because can precsely model both the ntra-class geometry and the nter-class dscrmnatve normaton n the local patch. 4. CONCUSION In ths paper we have proposed a novel lnear dmenson reducton algorthm termed Dscrmnatve essan Egenmaps (). s superor to the conventonal dmensonalty reducton algorthms because t ocuses on accurately modelng both the ntra-class geometry and nterclass dscrmnaton n the local patch. Emprcal studes on ace recognton tasks demonstrate that s more eectve than conventonal algorthms. 5. ACKNOWEDGEMEN S. S and K.-P. Chan thank the support rom KU-SPF Grant (project number 0006). D. ao thanks support rom the Nanyang SUG Grant (project number M52000) and Mcrosot Operatons PE D-NU Jont R&D (project number M420065). 6. REFERENCES []. otellng Analyss o a Comple o Statstcal Varables nto Prncpal Components Journal o Educatonal Psychology vol. 24 pp [2] R. A. Fsher he Use o Multple Measurements n aonomc Problems Annals o Eugencs vol. 7 pp [3] D. ao et al. General ensor Dscrmnant Analyss and Gabor Features or Gat Recognton IEEE rans. Pattern Analyss and Machne Intellgence vol. 29 no. 0 pp [4]. e and P. Nyog ocalty Preservng Projectons NIPS vol. 6 Vancouver Canada [5]. Zhang et al. Dscrmnatve ocalty Algnment Proc. European Con. Computer Vson vol. pp [6]. Zhang et al. Patch algnment or dmensonalty reducton IEEE rans. Knowledge and Data Engneerng [7] D.. Donoho and C. Grmes essan Egenmaps: new locally lnear embeddng technques or hgh-dmensonal data Proc. Natonal Academy o Arts and Scences vol. 00 pp [8]. Zhang et al. A unyng ramework or spectral analyss based dmensonalty reducton Int l Jont Con. Neural Networks pp [9]. Sm et al. he CMU Pose Illumnaton and Epresson (PIE) Database o uman Faces echncal Report CMU-RI- R-0-02 Carnege Mellon Unversty 200 [0] D. P. Bertsekas Constraned Optmzaton and agrange Multpler (Optmzaton and Neural Computaton Seres) Athena Scentc 996. [] D. ao et al. Geometrc Mean or Subspace Selecton IEEE rans. Pattern Analyss and Machne Intellgence vol. 3 no. 2 pp [2] D. u et al. Margnal Fsher Analyss and Its Varants or uman Gat Recognton and Content Based Image Retreval IEEE rans. Image Processng vol. 6 no. pp [3] W. u et al. ransductve Component Analyss Proc. IEEE Int l Con. Data Mnng pp