FACE RECOGNITION BASED ON OPTIMAL KERNEL MINIMAX PROBABILITY MACHINE

Transcription

1 Joural of heoretcal ad Appled Iformato echolog 8 th Feruar 03. Vol. 48 No JAI & LLS. All rghts reserved. ISSN: jatt.org E-ISSN: FACE RECOGNIION BASED ON OPIMAL KERNEL MINIMAX PROBABILIY MACHINE ZHIQIANG ZHOU, ZIQIANG WANG, 3 XIA SUN School of Iformato Scece ad Egeerg, Hea Uverst of echolog, Zhegzhou Cha E-mal: cha6o@6.com ABSRAC Face recogto has receved etesve atteto due to ts potetal applcatos ma felds. o effectvel deal th ths prolem, a ovel face recogto algorthm s proposed usg the optmal kerel mma proalt mache. he ke dea of the algorthm s as follos: Frst, the dscrmatve facal features are etracted th local fsher dscrmat aalss (LFDA. he, the mma proalt mache (MPM s eteded to ts olear couterpart usg optmal data-adaptve kerel fucto. Fall, the face mage s recogzed usg the optmal kerel MPM classfer the dscrmatve feature space. Epermetal results o three face dataases sho that the proposed algorthm performs much etter tha tradtoal face recogto algorthms. Keords: Face Recogto, Mma Proalt Mache, Feature Etracto, Kerel Fucto. INRODUCION I recet ears, face recogto has ee recevg more ad more atteto patter recogto ad computer vso felds. he motvato ehd face recogto s to emplo t to mplemet vdeo survellace, dett authetcato, ad huma-computer teracto. As a result, a large umer of face recogto algorthms have ee proposed, ad surves ths area ca e foud [].o ssues are cetral to all these algorthms: ho to etract dscrmatve facal features ad ho to classf a e face mage ased o the etracted facal features. herefore, ths ork also focuses o the to ssues of feature etracto ad classfer selecto. Prcpal compoet aalss (PCA ad lear dscrmat aalss (LDA are to classc feature etracto algorthms for face recogto []. he major dea of PCA s to decompose a data space to a lear comato of a small collecto of ases, hch are parse orthogoal ad capture the drectos of mamum varace the trag set. As a usupervsed feature etracto algorthm, PCA s optmal terms of represetato ad recostructo, ut ot for dscrmatg oe face class from others. LDA s a supervsed feature etracto algorthm hch ams to fd a projecto suspace o hch the data from the same class ll e pushed close hle the data from dfferet classes ll e pulled far aa. he projecto vectors are commol otaed mamzg the etee-class scatter ad smultaeousl mmzg the th-class scatter. Due to the utlzato of class lael formato, LDA s epermetall reported to outperform PCA for face recogto he suffcet laeled face mages are provded [3]. Hoever, the performaces of LDA are ofte degraded the lmted avalale dmesoal space ad the sgulart prolem. I addto, depedet compoet aalss (ICA s aother lear feature etracto algorthm [4], hch separates the hgh-order momets of the put data esdes the secod-order momets PCA. Hoever, the ojectve of ICA s to make the compoets of projected vectors as depedet as possle, hch ma ot ecessarl e the est for classfcato prolem such as face recogto. Although PCA ad LDA have del ee appled to mage retreval, face recogto, ad formato retreval, the ma fal to dscover the uderlg mafold structure as the seek ol a compact Eucldea suspace for face represetato ad recogto [5]. Follog the aove aalss, t s desred to propose a effcet algorthm for feature etracto eplctl cosderg the possl local mafold structure of face mage space. Local fsher dscrmat aalss (LFDA [6] s a recetl proposed mafold learg algorthm hch ams to mamze etee-class separalt ad preserve th-class local mafold structure at the same tme. hus LFDA s helpful for feature etracto of face mage data. 645

2 Joural of heoretcal ad Appled Iformato echolog 8 th Feruar 03. Vol. 48 No JAI & LLS. All rghts reserved. ISSN: jatt.org E-ISSN: As for face recogto, classfer selecto s aother ke ssue after feature etracto. At preset, the earest eghor (NN classfer s del used the face recogto algorthm, hch orks fdg the eghor th the mmum dstace etee the quer stace ad all laeled data staces. Although the NN classfer s the smplest oe for patter classfcato, ts performace deterorates dramatcall he the put data set has a relatvel lo local relevace. Support vector mache (SVM [7] s a popular patter classfcato method used recet ears. It otas top-level performace dfferet applcatos ecause of ts good geeralzato alt mmzg the VC dmeso ad achevg a mmal structural rsk. he asc dea ehd SVM s to fd a optmal hperplae a hgh dmesoal feature space that mamzes the marg of separato etee the closest trag eamples from dfferet classes. Although SVM classfer has acheved great success ma patter classfcato tasks, oe major draack of SVM s that t ca ot ota a eplct upper oud o the proalt of msclassfcato of future data. Recetl, mma proalt mache (MPM [8, 9] classfer has ee of de cocer sce t ca ota a eplct orst-case oud o the proalt of msclassfcato of future data. Hoever, MPM fals to cosder ho to select a optmal kerel fucto that adapts ell to the put data ad the learg. I ths paper, e propose a optmal adaptve kerel fucto to mamze the class separalt the kerel feature space. he fal optmzed kerel MPM shos that t s more adaptve to the face mage data ad leads to a sustatal mprovemets the performace of face recogto. he rest of the paper s orgazed as follos. I secto, e troduce ho to etract dscrmatve facal feature th LFDA. A effectve face recogto algorthm ased o optmal kerel MPM s proposed Secto 3. Epermetal results are sho Secto 4. Coclusos are reported Secto 5.. DISCRIMINAIVE FACIAL FEAURE EXRACION WIH LFDA Local fsher dscrmat aalss (LFDA [6] s a recetl proposed mafold learg algorthm for dscrmatve feature etracto, hch comes the advatages of LDA ad localt preservg projecto. It selects features through mamzg etee-class separalt ad preservg the th-class local mafold structure at the same tme, thus achevg mamum dscrmato. D Gve a set of face mages,,,, Let X [,,, ]. Let S ad S deote the local th-class scatter matr ad the local etee-class scatter matr, respectvel. her deftos are as follos: S Wj ( j( j (, j S Wj ( j( j (, j Where W ad W are to local eght matrces defed o the data pots, ther compoets ca e otaed through the follog computato: W W j j Aj, f c cj l l 0, f c cj A j, f c cj l l, f c cj (3 (4 Where c deotes the class lael of mage, l {,,, c} represet the class lael of mage, c deotes the total umer of class lael, ad l deotes the umer of mages the l th class. A j deotes the follog afft matr: j Aj ep σσ j (5 here σ represets the local scalg of the mages aroud, hch s determed σ (6 k k here s the k-earest eghor of mage, ad the parameter k s emprcall set to 7 the follog epermets. LFDA ams to fd a optmal trasformato D d matr U solvg the follog mamzato prolem: ( ( r U S U J( U arg ma (7 U r U S U 646

3 Joural of heoretcal ad Appled Iformato echolog 8 th Feruar 03. Vol. 48 No JAI & LLS. All rghts reserved. ISSN: jatt.org E-ISSN: As ca e see from the aove optmal ojectve fucto, LFDA looks for a optmal trasformato matr such that ear data pars the same class are made close ad the data pars dfferet classes are separated from each other; far apart data pars the same class are ot mposed to e close. hus, the optmal U are the egevectors assocated th the largest egevalues of the follog geeralzed ege-prolem: SU λ (8 SU Sce S s osgular after some preprocessg (such as PCA projecto steps o X, the colum vectors of U ca also e regarded as the egevectors of the matr S S assocated th the largest egevalues. Let the colum vectors U, U,, Ud e the soluto of (8 ordered accordg to ther egevalues λ > λ > > λ d. hus, the dscrmatve feature of mage ca e computed as follos: U (,,, U U U U d (9 here s the loer-dmesoal dscrmatve feature represetato of the face mage, ad U s the trasformato matr. No, e get the loer-dmesoal feature represetato of the orgal face mages. I the reduced sematc space, those face mages elogg to the same class are close to oe aother ad elogg to dfferet classes are far aa each other. herefore, e ca appl effectve classfer algorthm to mplemet face recogto the reduced feature space. I the et secto, e ll troduce ho to classf dfferet face mages th optmal kerel MPM classfer. 3. FACE RECOGNIION WIH OPIMAL KERNEL MPM CLASSIFIER Mma proalt mache (MPM [8, 9] classfer s a recetl proposed classfer algorthm. he ma advatage of MPM s that t ca mmze the orst-case proalt of msclassfcato of future data pots uder all possle choces of class-codtoal destes th gve mea ad covarace matr. I addto, t ca cope th the olear decso oudares eplotg kerel trck. I the follog, e frst troduce MPM ad ts kerel eteso, ad the e dscuss ho to select a optmal kerel fucto that adapts ell to the put face mage data ad the face recogto task, so that the recogto accurac ca e guarateed. Let ad deote to facal feature vectors the reduced feature space, ad ther meas vectors, ad covarace matrces are represeted as { } ad {, }, respectvel. MPM ams to fd the m hperplae a z a \ 0, ad z hch ca separate the to classes of pots th mamal proalt th respect to ther meas ad covarace matrces. It ca e formall descred as follos: α, a, ( { } { } { a } ma α st.. α sup Pr a α sup Pr (0 here α represets the loer oud of the accurac for the classfcato of future data pots. B usg the follog theorem troduced [8]: sup Pr{ a } + d th d f Σ a ( he prolem (0 ca e equvaletl trasformed to the follog prolem: m a Σ a + a Σ a s.t. a ( a hs s a secod order coe program (SOCP prolem, hch ca e solved usg teror methods [0]. Oce the optmal a ad are otaed solvg ( ad (0, the classfcato of e data pot z s doe computg: sg +, elogs to class z (3 otherse, z elogs to class ( a z he MPM algorthm descred aove s a lear method, hch ma fal to deal th hghl olear data. o eted MPM to the olear case, e dscuss ho to perform MPM Reproducg Kerel Hlert Space (RKHS [7], hch gves rse to kerel MPM. Let face mage data ad e mapped to kerel space va olear mappg fucto ϕ : ϕ, ϕ ϕ (4 647

4 Joural of heoretcal ad Appled Iformato echolog 8 th Feruar 03. Vol. 48 No JAI & LLS. All rghts reserved. ISSN: jatt.org E-ISSN: ϕ, ϕ ϕ (5 Kerel MPM ams to fd the hperplae a ϕ z hch ca separate the to classes of pots th mamal proalt th respect to ther meas ad covarace matrces the kerel space. Smlar to MPM, the optmal ojectve fucto of kerel MPM ca e descred as follos: m a a Σ a + a Σ a ϕ ϕ ( ϕ ϕ s.t. a (6 Sce a soluto a must le the spa of all the samples the kerel feature space, there ests,,, ad j,,, such that αϕ βϕ (7 a + j Susttutg (7 to (6 ad usg the kerel fucto K( z, z ϕ( z ϕ( z, the optmal prolem (6 ca e rertte as here m γ γ K K γ + γ K K γ ( k k s.t. γ (8 teror-pot methods. Oce the optmal γ s otaed, the classfcato decso rule of kerel MPM s gve sg ϕ + sg [ γ ] K( z, z f z a z ( (4 If f ( z + the the testg data z s classfed as from class, otherse the testg data z s classfed as from class. From the aove computer process, e ca oserve that kerel fucto K pla a mportat role kerel MPM. he most commol used kerels clude Gaussa kerel ad polomal kerel. Hoever, the olear structure captured these data-depedet kerels ma ot e cosstet th the trsc mafold structure. o mprove the classfcato performace of kerel MPM algorthm, e adopt the follog adaptve kerel learg method. Sce the data-depedet kerel ca e otaed va parse costrats [], e ca costruct the follog smlart matr to represet the parse costrats ( j +,, SP j,, j DP 0, otherse (5,,,,,,, γ α α α β β β (9 (0 + k th k K(, j j z ( + k th k K(, j j z, for,,, z for,,, K k K K K K k ( (3 m s a colum vector th oes of dmeso m, K ad K deote the frst ros ad ros of the kerel matr K. Sce the optmal prolem (8 s also a secod order coe program, hch ca e solved usg Where SP deotes smlar parse costrat (the data pars share the same class, ad DP deotes dssmlar parse costrat (the data pars have dfferet classes. Let L e the ormalzed graph Laplaca matr defed as follos: L I D WD (6 here I s the dett matr, D s a dagoal matr th ts elemets are D W j, ad W s the eght matr defed o the hole data set, ts defto s as follos:, f s amog the k earest eghor of j W or j s amog the k erest eghor of (7 j 0, otherse he the mafold adaptve kerel learg ca e formulated the follog mmzato prolem: m r K 0 ( LK C l ( j Kj + (, j S D j (8 648

5 Joural of heoretcal ad Appled Iformato echolog 8 th Feruar 03. Vol. 48 No JAI & LLS. All rghts reserved. ISSN: jatt.org E-ISSN: here r s the matr trace, l s the square hge loss fucto, ad C s the postve costat to cotrol the tradeoff etee the emprcal loss l ad the trsc data mafold. Sce the optmal prolem (8 elogs to tpcal sem-defte programmg (SDP prolem, hch ca e easl computed usg the stadard SDP solver SeDuM []. B usg the otaed optmal kerel fucto K the computato process of kerel MPM, e ca greatl mprove the classfcato performace of the kerel MPM algorthm. he latter epermetal results valdate ths cocluso. I short, our proposed face recogto algorthm has to steps. Frst, e etract the dscrmatve facal feature th LFDA, ad the the face mage s recogzed usg the optmal kerel MPM classfer the reduced feature space. 4. EXPERIMENAL RESULS I ths secto, e evaluate the performace of our proposed algorthm ad compare t th the state-of-the-art algorthms for face recogto. Our emprcal stud o the face recogto as coducted ased o three real-orld face dataases. he facal recogto techolog (FERE face dataase [3] s a commol used dataase for the test of state-of-art face recogto algorthms. I the follog, the proposed algorthm s tested o a suset of ths dataase hch cotas 400 mages of 00 sujects. he suset cotas the mages hose ames are marked th to-character strgs: a, j, k, e, f, d, ad g. Each suject has seve mages volvg varatos llumato, pose, ad facal epresso. I our epermet, each orgal mage s cropped so that each cropped mage ol cotas the portos of the face ad har. he, the facal areas ere cropped ad reszed to 3 3 ad preprocessed the hstogram equalzato. Some sample mages after preprocessg of the FERE dataases are sho Fgure. S out of seve mages of each suject are radoml chose for trag, ad the remag oe s used for testg. hus, the trag set sze s 00 ad the test set sze s 00.he fal recogto accurac s computed averagg all te trals. he UMIS dataase cotas 0 persos th totall 564 face mages [4]. here are varatos of race, se, ad appearace th dfferet sujects. he sze of each mage s appromatel 0 0 pels, th 56 gra levels per pel, hch are 649 reszed to 3 3 pels epermet. I addto, preprocessg to locate the faces as appled. Orgal mages ere ormalzed such that the to ees ere alged at the same posto. Some sample mages after preprocessg of the UMIS dataases are sho Fgure.he trag set s a radoml selected suset th te mages per dvdual, ad the remag mages of the dataase are used as the testg set. hus, the trag set sze s 00.he fal recogto accurac s computed averagg all te trals. he Yale face dataase ( projects/alefaces/alefaces.html cotas 65 gra scale mages of 5 dvduals. he mages demostrate varatos lghtg codto, facal epresso. I ths epermet, all the mages are alged fg the locatos of the to ees. Hstogram equalzato s appled as a preprocessg step. Some sample mages after preprocessg of the Yale dataases are sho Fgure 3. We radoml select fve mages of each dvdual to costruct the trag set ad the rest mages of the dataase to form the testg set. hus, the umers of the trag samples ad testg samples are 75 ad 90, respectvel. he fal recogto accurac s computed averagg all te trals. Fgure : Face Image Eamples from the FERE Dataase Fgure : Face Image Eamples from the UMIS Dataase Fgure 3: Face Image Eamples from the Yale Dataase o evaluate our proposed optmal kerel MPM (OKMPM algorthm, e sstematcall compare t th Egeface [], Fsherface [], Laplacaface [5], SVM [7], ad the orgal MPM [9] algorthms o FERE, UMIS, ad Yale dataases. For far comparso, e frst appl LFDA to etract facal feature, the SVM (or MPM classfer s adopted to recogze dfferet face mages the reduced feature space. he classfcato accurac for each algorthm o the three dataases s reported o the

6 Joural of heoretcal ad Appled Iformato echolog 8 th Feruar 03. Vol. 48 No JAI & LLS. All rghts reserved. ISSN: jatt.org E-ISSN: ale -ale 3, respectvel. From these results, e ca make the follo oservatos: our proposed OKMPM performs much etter tha Egeface, Fsherface, ad Laplacaface, SVM, ad MPM algorthms o the three dataases, hch sho that smultaeousl usg the LFDA-ased feature etracto method ad the optmal kerel MPM classfer ca effectvel mprove the performace of face recogto. Egeface performs the orst. Laplacaface outperforms Egeface ad Fsherface sce Laplacaface cosders the mafold structure of face mages. 3 Although MPM acheves etter performace tha SVM eplctl cosderg the loer oud of the classfcato accurac, t stll performs orse tha our proposed OKMPM. he ma reaso could e attruted to the fact that OKMPM ca effectvel capture the olear mafold structure th the optmal data-adaptve kerel fucto. ale : Recogto Accurac Comparsos O he FERE Dataase Algorthm Accurac Egeface 6.% Fsherface 68.5% Laplacaface 79.3% SVM 84.6% MPM 85.3% OKMPM 89.4% I addto, to test hether our proposed optmal adaptve kerel reall mprove the performace of kerel MPM, e also test the performace of kerel MPM he the kerel s set dfferet kerel fuctos, such as Gaussa kerel, polomal kerel, ad Sgmod kerel. he epermetal results o the three dataases are sho ale 4. As ca e see, our proposed optmal kerel fucto acheves the est performace amog the compared kerel fuctos. he possle eplaatos are as follos: Gaussa kerel, polomal kerel, ad sgmod kerel are all the data-depedet kerels, hch ma ot e cosstet th trsc mafold structure. Hoever, our proposed optmal kerel s otaed usg the parse costrats ad eplotg the local geometr of face mages. herefore, the otaed optmal kerel ca effectvel capture the olear mafold structure of face mages, hch leads to etter performace of kerel MPM. ale : Recogto Accurac Comparsos O he UMIS Dataase Algorthm Accurac Egeface 9.8% Fsherface 93.% Laplacaface 94.4% SVM 95.8% MPM 96.% OKMPM 98.9% ale 3: Recogto Accurac Comparsos O he Yale Dataase Algorthm Accurac Egeface 56.% Fsherface 77.6% Laplacaface 88.4% SVM 90.3% MPM 90.7% OKMPM 93.5% ale 4: Recogto Accurac Of Kerel MPM Comparsos Uder Dfferet Kerel Fuctos Kerel FERE UMIS Yale Gaussa kerel Polomal kerel Sgmod kerel Optmal kerel 5. CONCLUSION 86.4% 94.3% 89.5% 86.% 93.9% 89.% 85.6% 93.7% 89.0% 89.4% 98.9% 93.5% I ths paper, e have proposed a ovel face recogto algorthm ased o the optmal kerel mma proalt mache (OKMPM. It ca effectvel capture the olear mafold structure th the optmal data-adaptve kerel fucto ad ota a eplct upper oud o the proalt of msclassfcato of future data. he epermetal results sho that the proposed OKMPM algorthm performs much etter tha tradtoal face recogto algorthms. I our future ork, e ll 650

7 Joural of heoretcal ad Appled Iformato echolog 8 th Feruar 03. Vol. 48 No JAI & LLS. All rghts reserved. ISSN: jatt.org E-ISSN: focus o the theoretcal aalss ad acceleratg ssues of our OKMPM algorthm. ACKNOWLEDGMENS hs ork s supported NSFC (Grat No , the Natoal Scece Foudato for Post-doctoral Scetsts of Cha (Grat No. 0M500035, ad the Specalzed Research Fud for the Doctoral Program of Hgher Educato of Cha (Grat No REFRENCES: [] W. Zhao, R. Chellappa, P. Phllps, ad A. Rosefeld, Face recogto: a lterature surve, ACM Computg Surves, Vol.35, No.4, 003, pp [] P.Belhumeur, J. Hespaha, ad D. Kregma, Egefaces vs. fsherfaces: recogto usg class specfc lear projecto, IEEE rasactos o Patter Aalss ad Mache Itellgece, Vol.9, No.7, 997, pp [3] Z.Fa, Y.Xu, ad D.Zhag, Local lear dscrmat aalss frameork usg sample eghors, IEEE rasactos o Neural Netorks, Vol., No.7, 0, pp.9-3. [4] D.V.Maarte, N.Dmtr, ad V.H.Sae, A comato of parallel factor ad depedet compoet aalss, Sgal Processg, Vol.9, No., 0, pp [5] X.He, S.Ya, Y.Hu, P.Nog, ad H.-J. Zhag, Face recogto usg Laplacafaces, IEEE rasactos o Patter Aalss ad Mache Itellgece, Vol.7, No.3, 005, pp [6] M.Sugama, Dmesoalt reducto of multmodal laeled data local fsher dscrmat aalss, Joural of Mache Learg Research, Vol.8, 007, pp [7] K.-R.Muller, S.Mka, G.Ratsch, K.suda, ad B.Scholkopf, A troducto to kerel-ased learg algorthms, IEEE rasactos o Neural Netorks, Vol., No., 00, pp.8-0. [8] G.R.G. Lackret, L.E.Ghaou, C.Bhattachara, ad M.I.Jorda, A roust mma approach to classfcato, he Joural of Mache Learg Research, Vol.3, 00, pp [9] G.R.G.Lackret, L.E.Ghaou, C.Bhattachara, ad M.I.Jorda, Mma proalt mache, Advaces Neural Iformato Processg Sstems, Vol.4, 00, pp [0] I.W H. sag, J..-Y. Kok, Effcet hperkerel learg usg secod-order coe programmg, IEEE rasactos o Neural Netorks, Vol.7, No., 006, pp [] J. Zhuag, I.W.sag, ad S.C.H.Ho, SmpleNPKL: smple o-parametrc kerel learg, Proceedgs of the 6th Iteratoal Coferece o Mache Learg (ICML, 009, pp [] B.Borchers, CSDP, a C lrar for semdefte programmg, Optmzato Methods & Softare, Vol., No., 999, pp [3] P.J.Phllps, M.Heojoo, S.A.Rzv, ad P.J.Rauss, he FERE evaluato methodolog for face-recogto algorthms, IEEE rasactos o Patter Aalss ad Mache Itellgece, Vol., No.0, 000, pp [4] D.B.Graham ad N. M. Allso, Characterzg vrtual egesgatures for geeral purpose face recogto, NAOASI Seres F, Computer ad Sstems Sceces, Vol.63, 998, pp