A New Regularized Orthogonal Local Fisher Discriminant Analysis for Image Feature Extraction

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1 ZHONGFENG WANG et al: A NEW REGULARIZED ORHOGONAL LOCAL FISHER DISCRIMINAN A Ne Regulazed Othogonal Local Fshe Dscmnant Analyss fo Image Featue Extacton Zhongfeng WANG, Zhan WANG 2 dept. name of oganzaton, name of oganzaton, Cty, County. School of Management Engneeng, Henan Insttute of Engneeng, Zhengzhou 459, P.R. Chna 2. Insttute of Infomaton Scence, Beng Jaotong Unvesty, Beng 00044, P.R. Chna Astact Local Fshe Dscmnant Analyss (LFDA) s a featue extacton method hch comnes the deas of Fshe dscmnant analyss (FDA) and localty pesevng poecton (LPP). It oks ell fo multmodal polems. But LFDA suffes fom the unde-sampled polem of the lnea dscmnant analyss (LDA). o deal th ths polem, e popose a egulazed othogonal local Fshe dscmnant analyss (ROLFDA) to mpove the pefomance of LFDA. ROLFDA fnds the optmal poecton vectos n the ange space of local mxtue scatte matx. he ne local thn-class scatte matx s appoxmated y egulazaton method n ode to egulate the egenvalues of the nely local thn-class scatte matx. he poecton vectos ae othogonal y utlzng the tace ato cteon. Expemental esults on the Yale, ORL and CMU PIE face dataases demonstate that the ROLFDA s an effectve algothm. Keyods - Infomaton extacton and patten analyss; Fshe dscmant analyss; mage featue extacton, undesampled polem; face dataases I. INRODUCION Pncpal component analyss (PCA) [] and lnea dscmnant analyss (LDA) [2] ae to famous dmensonalty educton algothms and have een dely used n the aeas of patten ecognton and compute vson. PCA ams to fnd a set of mutually othogonal ass vectos y maxmzng the vaance of the poected data. Rathe than PCA, LDA s a supevsed technque. It s to fnd an optmal poecton matx y maxmzng the ato of the eteen-class scatte to the thn-class scatte. PCA and LDA can only dscove the gloal Eucldean stuctue of the data. hey fal to fnd the nonlnea data stuctue. Recent studes sho that the face mages may esde on a nonlnea su-manfold [3]. Manfold-ased leanng algothms have een poposed to dscove the ntnsc lodmensonal stuctue. LPP [3] as poposed fo face ecognton staghtly. It fnds an emeddng suspace that pesevng local nfomaton and dscoveng the essental face manfold stuctue. Fo face ecognton o classfcaton, dscmnant nfomaton s vey mpotant. LDA-ased methods have een poposed to tackle the polem of tadton LDA, such as the undesampled polem. LPP also can e extended to supevsed and dscmnant models. Recently, Sugyama poposed local fshe dscmnant analyss (LFDA) [4], hch comnes the deas of FDA and LPP. It efomulates LDA and utlzes the neghohood nfomaton ndect to edefne the local eteen-class scatte matx and the local thn-class scatte matx. Can ok ell hen thn-class multmodalty o outles exst. LFDA also suffes fom the undesampled polem as ell as LDA. o ovecome ths daack, e popose a egulazed othogonal local fshe dscmnant analyss (ROLFDA) method to mpove the pefomance of LFDA. We fstly fnd the poecton matx n the ange space of local mxtue scatte matx. he nely local thn-class scatte matx can e appoxmated ell y the egulazaton method [5,6] n ode to allevate the ad mpacts of small and zeo egenvalues caused y nose and lmted nume of tanng samples. he tace ato cteon as appled to otan the othogonal poecton vectos. Snce the othogonal poecton s of desale popety and often has good pefomance empcally. he poposed method can mpove the pefomance of LFDA sgnfcantly. Expement esults on some face dataases vefy the effectveness of the poposed algothm. he est of ths pape s oganzed as follos. In secton 2, e eve the dea of local fshe dscmnant analyss (LFDA). In secton 3, e popose a egulazed othogonal local fshe dscmnant analyss (ROLFDA) n detal. In secton 4, expements th face mages dataases ae mplemented to demonstate the effectveness of ROLFDA algothm. Conclusons ae gven n secton 5. II. OVERVIEW OF LFDA LFDA s also a dscmnant analyss method. Gven N Samples X x, x2, xn n n -dmensonal space.he class lael of sample x s denoted as y, 2, c. Nc s the nume of classes. We denote n c as the nume of the samples n the c th class. LFDA edefnes the eteen-class metc matx S and the th-class metc matx S as follos: DOI 0.503/IJSSS.a ISSN: x onlne, pnt

2 ZHONGFENG WANG et al: A NEW REGULARIZED ORHOGONAL LOCAL FISHER DISCRIMINAN () S W ( x x ) ( x x ) N N ( ) 2 S W ( x x ) ( x x ) N N ( ) 2 (2) ( ) ( ) W and W s the eght matces: ( ) f ( ) A n nl y y W n f y y (3) f ( ) A nl y y W 0 f y y (4) A s the affnty matx, A eflects the elatonshp 2 eteen x and x. A exp( x x / ). ( k ) ( ) x x. x k s the k -th neaest negho of x. If the ente element of A ae one, the LFDA method can e tansfomed to standad LDA. he oecton functon of LFDA s: V SV t( V SV) Vopt ag max ag max (5) ( V V S V V t V SV ) he aove optmal polem s tansfomed to genealzed egenvalue polem as follos: Sv Sv,,2,, d (6) V [ v,, v d ] ae the egenvectos coespondng to the d lagest egenvalues. Because of the choce of A, the LFDA can eveal some local nfomaton and ovecome some daacks of tadton LDA. Let St S S e the local mxtue scatte matx, then the optmal polem (5) s equvalent to: III. V V SV V SV ag max ag max V V ( S S ) V V V SV opt t REGULARIZED ORHOGONALLOCAL FISHER DISCRIMINAN ANALYSIS A. Dscmnant featue analyss We decompose the hole space nto the follong fou sunspaces accodng to the egenvalues of S and S [7]. Suspace : span ( S ) span( S ). In ths suspace, the egenvalues of S and S ae geate than zeo. he dscmnant nfomaton exsts n ths suspace. Suspace 2: span ( S ) null( S ). In ths suspace, the egenvalues of S ae geate than zeo, the egenvalues of S ae equal zeo. he dscmnant nfomaton exsts n ths suspace. (7) Suspace 3: null ( S ) span( S ). In ths suspace, the egenvalues of S ae equal zeo, the egenvalues of S ae geate than zeo. he dscmnant nfomaton exsts n ths suspace. Suspace 4: null ( S ) null( S ). In ths suspace, the egenvalues of S and S ae equal zeo. he oectve functon (5) s meanngless. hs suspace does not contan dscmnant nfomaton. Snce the null space of the local mxtue scatte matx s the ntesecton of the null space of S and the null space of S. We can emove the null space of St thout dscadng dscmnant nfomaton. Suppose that the sngula value decomposton (SVD) of St s St U Q (8) Snce the matx St s symmetc and postve semdefnte, U= Q. If the ank of St s, U s the matx consstng of the fst columns of matx U. So the cost functon (5) s equvalent to the follong oecton functon y V U P: PU SUP P SP Popt ag max ag max P P PU SUP P SP (9) hee S U SU, S U SU. he othogonal constant s added to avod the tval soluton. So the oecton functon (9) s changed to: P SP Popt ag max (0) P PI P S P B. Regulazed othogonal local fshe dscmnant analyss In fomula (0), S may e sngula. Fo allevatng the mpacts of small and zeo egenvalues, e utlzed egulazaton technques to egulate the egenvalues of S. S s symmetc and postve semdefnte, So the sngula value decomposton of matx S s epesented as follos: m t S u u u u () hee 2 t t m 0, t s the ank of matx S. As ponted n [5], the S also can e decomposed nto thee suspaces: the face suspace, nose suspace, null suspace. So the S can e expessed as follos: DOI 0.503/IJSSS.a ISSN: x onlne, pnt

3 ZHONGFENG WANG et al: A NEW REGULARIZED ORHOGONAL LOCAL FISHER DISCRIMINAN k t m S u u u u u u k t face suspace nose suspace null suspace (2) In LPP, PCA as used fo dmensonalty educton, keep 98% nfomaton and get the pncpal components. So the value of k s calculated y the follong ule: k 98% (3) he functon fom f [5] as used to model the spectum of face space: k t m S uu uu uu k t (4) s a constant,, ae paametes and can e detemned y lettng, k,and get k k( k ) k k, k k (5) S s egaded as the appoxmaton of S, Let S (6) S P opt, the optmal polem (0) can e changed nto: P SP t( P SP) ag max ag max P PI P S P P I P t( P S P) C. Fndng the optmal soluton IV. EXPERIMENS o evaluate the poposed ROLFDA algothm, e compaed t th PCA, LDA, LPP, LFDA and egenfeatue egulazaton and extacton (ERE) on Yale, ORL and CMU PIE face dataases. LDA, LPP and LFDA need PCA pepocessng to avod the sngulaty polem and 98% enegy as kept n PCA phase. he neaest neghohood classfe th Eucldean dstance metc as used n expements. A. Expement on Yale face dataase he Yale face dataase [] contans 65 gayscale mages of 5 ndvduals. Each ndvdual has mages. hese mages ee captaned unde vaous lghtng condton (cente-lght, left-lght, ght-lght), facal expessons (nomal, happy, sad, supse and nk) and facal detals (th glasses o thout). he ognal sze of the mages n ths dataase s pxels. In ou expements, all mages ee copped nto 3232 y fxng the locaton of the to eyes. We andomly selected ll ( 3,4,5,6) mages fom each ndvdual fo tanng, hle the est mages ee used fo testng. Fo each gven l, e epeated expements ove 50 andom splts and the aveage ecognton ates ee otaned. Fo LDA and ERE, the dmensonaltes of featue space ae at most 4. Fo LFDA and ROLFDA, the paametes ae set to l. he paamete n fomula (4) s chosen as one. he est pefomances of dffeent methods th coespondng dmensonalty ee pesented n ale. Fom ale, ROLFDA acheves hghe accuacy than the othe methods. Next, e expemented th dffeent dmensonaltes of featue space. A andom suset th 5 mages pe ndvdual as used fo tanng, and the emanng mages ee used fo testng. he aveage ecognton ates ove 50 ndependent uns ae shon n Fg.(). he fomula (6) s a tace ato polem. he tace ato polem does not have a closed-fom soluton. It s usually appoxmated y solvng a ato tace polem ag max t[( P S P) ( P S P)]. Hoeve, thee P PI exst dffeent methods to solve the tace ato polem. Wang et.al poposed an teatve method called teatve tace ato (IR) to solve the tace ato polem [8]. Yang.et.al poposed a moe effcent algothm called decomposed Neton s method (DNM) to optmze the tace ato polem [9]. In IR algothm, the poecton matx s ntalzed as an atay column othogonal matx. DNM ntalzes the tace ato value as zeo, and the update ule s changed. Zhao.et.al gave the loe ounday of tace ato value [0]. Fo polem (6), e can otan that the loe ounday of tace ato value s t( S)/ t( S ). So e apply the DNM method to solve the tace ato polem (6) th the fxed ntalzaton of tace ato value. Fgue. Recognton Rates (%) Vesus the Dmenson of Featue Space on the Yale Face Dataase DOI 0.503/IJSSS.a ISSN: x onlne, pnt

4 ZHONGFENG WANG et al: A NEW REGULARIZED ORHOGONAL LOCAL FISHER DISCRIMINAN ABLE I. HE OP RECOGNIION RAE (%) OF DIFFEREN MEHODS ON HE YALE FACE DAABASE Method 3 tan 4tan 5tan 6tan PCA 5.0(40) 54.86(60) 58.09(74) 60.72(89) LDA 60.70(4) 68.59(4) 74.3(4) 77.55(4) LPP 62.3(4) 68.97(4) 74.04(4) 77.36(4) LFDA 62.98(4) 69.85(4) 74.3(4) 77.55(4) ERE 64.58(4) 72.25(4) 76.69(4) 78.08(4) ROLFDA 66.62(5) 74.29(9) 78.7(22) 82.48(7) B. Expement on ORL face dataase he ORL dataase [2] contans 400 facal mages of 40 ndvduals. Each ndvdual has 0 dffeent mages. All the mages ee taken aganst a dak homogeneous ackgound th the suects n an upght, font poston (th toleance fo some sde movement). he ognal sze of the mages s 292 pxels. In ou expement, the mages ee manually copped and eszed to 3232 pxels. C. Expement on CMU PIE face dataase he CMU PIE face dataase [3] contans 68 people th 4368 face mages. he face mages ee captaned unde vayng pose llumnaton and expesson. We selected a suset (C29) hch contans 632 mages of 68 ndvduals (each people has 24 mages). he mages ee manually copped and eszed 3232 pxels. We andomly selected ll ( 4,8,2,6) mages fom each ndvdual fo tanng, hle the est mages ee used fo testng. Fo each gven l, 20 ndependent uns ee pefomed and the aveage ecognton ates ee otaned. he est pefomances of dffeent methods th coespondng dmensonalty ee pesented n ale 3. Fom ale 3, ROLFDA outpefoms othe methods. ABLE III. HE OP RECOGNIION RAE (%) OF DIFFEREN MEHODS ON HE CMU PIE FACE DAABASE Method 4 tan 8tan 2tan 6tan PCA 4.8(99) 63.96(48) 75.25(97) 77.22(278) LDA 8.95(67) 89.34(67) 89.77(67) 88.69(67) LPP 82.54(44) 89.(67) 90.0(67) 90.69(75) LFDA 82.87(99) 89.89(92) 90.53(94) 90.7(87) ERE 83.20(67) 90.53(67) 9.54(67) 92.03(67) ROLFDA 84.5(67) 90.99(77) 9.9(82) 93.0(97) Fgue 2. Recognton Rates (%) Vesus the Paamete on the ORL Face Dataase. ABLE II. HE OP RECOGNIION RAE (%) OF DIFFEREN MEHODS ON HE ORL FACE DAABASE Method 3 tan 4tan 5tan 6tan PCA 77.95(78) 83.92(00) 87.39(6) 90.30(48) LDA 86.3(39) 9.23(39) 94.0(39) 95.25(39) LPP 87.44(39) 9.05(39) 94.40(44) 95.72(42) LFDA 87.66(39) 9.20(39) 94.34(39) 95.63(43) ERE 90.04(39) 93.95(39) 96.50(39) 97.25(38) ROLFDA 9.(50) 94.96(42) 96.94(48) 97.56(44) We andomly selected ll ( 3,4,5,6) mages fom each ndvdual fo tanng, hle the est mages ee used fo testng. Fo each gven l, e epeated expements ove 40 andom splts and the aveage ecognton ates ee otaned. he est pefomances of dffeent methods th coespondng dmensonalty ee pesented n ale 2. Fom ale 2, ROLFDA s est. Next, e am to seach fo optmal n fomula (4). Fg.(2). shos the dffeent th coespondng ecognton ate. V. CONCLUSION In ths pape, e pesented a egulazed othogonal local Fshe dscmnant analyss (ROLFDA) fo dmensonalty educton and featue extacton. ROLFDA fnds the optmal soluton n the ange space of local mxtue scatte matx. he nely local thn-class scatte matx s appoxmated y egulazaton method n ode to allevate the unelale mpacts of small and zeo egenvalues hch caused y nose and lmted tanng samples. he tace ato cteon s appled to otan othogonal poecton vectos. Expemental esults on Yale, ORL and CMU PIE face dataases sho the effectveness of the poposed ROLFDA algothm. ACKNOWLEDGMEN hs ok as fnancally suppoted y Doctoal Fund Poect of Henan Unvesty of Engneeng (D205030). REFERENCES [] [] M.A. uk, A.P. Pentland, Egenfaces fo ecognton, J. Cogntve Neuosc. 3 () (99) [2] [2] P.N. Belhumeu, J.P. Hespanha, D.J. Kegman, Egenface vs. Fsheface: ecognton usng class specfc lnea poecton, IEEE ans. Patten Anal. Mach. Intell. 9 (7) (997) [3] [3] X. He, S. Yan, Y. Hu, P. Nyog, H. Zhang, Face ecognton usng Laplacanfaces, IEEE ans. Patten Anal. Mach. Intell. 27 (3) (2005) [4] [4] M. Sugyama, Dmensonalty educton of multmodal laeled data y local Fshe dscmnant analyss, J. Mach. Lean. Res. 8 (2007) [5] [5] X. Jang, B. Mandal, and A. Kot, Egenfeatue egulazaton and extac-ton n face ecognton, IEEE ans. Patten Anal. Mach. Intell., vol. 30,no. 3, pp , Ma DOI 0.503/IJSSS.a ISSN: x onlne, pnt

5 ZHONGFENG WANG et al: A NEW REGULARIZED ORHOGONAL LOCAL FISHER DISCRIMINAN [6] [6] J.Lu, Y P. an. Regulazed localty pesevng poectons and ts extensons fo face ecognton. IEEE ans Syst Man Cyen B Cyen. 200 Jun;40(3): [7] [7]Z S. Zhang,. Sm Dscmnant suspace analyss: a Fukunaga Koontz appoach IEEE ans. Patten Anal. Mach. Intell, 29 (0) (2007), pp [8] [8] H. Wang, S. Yan, D. Xu, and X. ang. ace ato vs. ato tace fo dmensonalty educton. In CVPR 07. IEEE Confeence on Compute Vson and Patten Recognton, [9] [9] Y. Ja, F. Ne, and C. Zhang, ace ato polem evsted, IEEE ans. Neual Net., vol. 20, no. 4, pp , Ap [0] [0] M.B Zhao, Z. Zhang, W. S. ommy: ace ato cteon ased genealzed dscmnatve leanng fo sem-supevsed dmensonalty educton. Patten Recognton 45(4): (202) [] []F. Samaa, A. Hate, Paametesaton of a stochastc model fo human face dentfcaton, n: Second IEEE Wokshop on Applcatons of Compute Vson, 994. [2] [2]Yale Unvesty Face Dataase, yalefaces.html, [3] [3]. Sm, S. Bake, and M. Bsat, he CMU pose, llumnaton, and ex-pesson (PIE) dataase of human faces Root. Instt., Canege- Mellon Unv., Pttsugh, PA, 200, ech. Rep. CMU-RI-R-0-02 DOI 0.503/IJSSS.a ISSN: x onlne, pnt