FACIAL IMAGE FEATURE EXTRACTION USING SUPPORT VECTOR MACHINES

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1 FACIAL IMAGE FEATURE EXTRACTION USING SUPPORT VECTOR MACHINES H. Abrsham Moghaddam K. N. Toos Unversy of Technology, P.O. Box , Tehran, Iran M. Ghayoum Islamc Azad Unversy, Scence and Research Un, P.O. Box , Tehran, Iran Keyords: Absrac: Feaure exracon, Suppor vecor machnes, Face recognon, Prncpal componen analyss, Independen componens analyss, Lnear dscrmnan analyss. In hs paper, e presen an approach ha unfes sub-space feaure exracon and suppor vecor classfcaon for face recognon. Lnear dscrmnan, ndependen componen and prncpal componen analyses are used for dmensonaly reducon pror o nroducng feaure vecors o a suppor vecor machne. The performance of he developed mehods n reducng classfcaon error and provdng beer generalzaon for hgh dmensonal face recognon applcaon s demonsraed. INTRODUCTION Choosng an approprae se of feaures s crcal hen desgnng paern classfcaon sysems under he frame-ork of supervsed learnng. Ideally, e ould lke o use only feaures havng hgh separably poer hle gnorng or payng less aenon o he res. Recenly, here has been an ncreased neres n deployng feaure selecon n applcaons such as face and gesure recognon (Sun e al., 004). Mos effors n he leraure have been focused manly on developng feaure exracon mehods (Jan e al., 000, Belhumeur, e al., 997) and employng poerful classfers such as probablsc (Moghaddam, 00), hdden Markov models (HMMs) (Ohman and Aboulnasr, 003), neural neorks (NNs) (Er e al., 00) and suppor vecor machne (SVM) (Lee e al., 00). The man rend n feaure exracon has been represenng he daa n a loer dmensonal space compued hrough a lnear or non-lnear ransformaon sasfyng ceran properes. Prncpal componen analyss (PCA) (Turk and Penland, 99) selecs feaures hch are maxmally varan across he daa. Wh ndependen componens analyss (ICA) (Lu and Wechsler, 003) sascally ndependen feaures resul. Lnear dscrmnan analyss (LDA) (Yu and Yang, 00) encodes he dscrmnaory nformaon n a lnear separable space by maxmzng he rao of beeen-class o hn-class varances. SVM have shon o be very effecve classfers for face recognon applcaons and provde he ably o generalze over magng varans (Hesele e al., 00). SVM provde an opmal decson hyperplane by employng kernel learnng, projecng he daa no a hgh-dmensonal space (Vapnk, 995). Some auhors used PCA and ICA for dmensonaly reducon before usng SVM for face recognon (Wang e al., 00, Q, e al., 00). Whou usng effecve schemes o selec an approprae subse of feaures n he compued subspaces, hese mehods rely mosly on classfcaon algorhms o deal h he ssues of redundan and rrelevan feaures. Ths mgh be problemac, especally hen he number of ranng examples s small compared o he number of feaures. Foruna and Capson (Foruna and Capson, 004) proposed an erave componen algorhm for feaure selecon by combnng PCA and ICA mehods and SVM classfer. In hs paper, e presen an approach ha uses SVM o classfy PCA, ICA and LDA exraced feaures and a hybrd erave mehod for mprovng he generalzaon of he classfer. Applcaon of he developed algorhm o a facal mage daabase

2 demonsraes he mprovemen n correcness, margn and number of suppor vecors of he classfer. The res of he paper s organzed as follos: Secon provdes a bref reve of feaure exracon algorhms ncludng PCA, ICA and LDA. In secon 3, e presen he classfcaon algorhm usng SVM and he erave mehod for mprovng he generalzaon of he classfer. Secon 4 s devoed o expermenal resuls and dscusson. Fnally, concludng remarks and plans for fuure orks are gven n secon 5. FEATURE EXTRACTION Gven a se of cenred npu vecors x, x,..., x N of n varables, a daa marx X s defned h each vecor formng a column of X. The goal of feaure exracon algorhms s o consruc a decomposon of he daa such ha a se of bass vecors for he daa hch are maxmally decorrelaed can be found. In oher ords, e look for a marx A such ha: S = A X () here he columns of S are decorrelaed. For paern recognon, he decorrelaed space S s used for dmensonaly reducon.. Prncpal Componen Analyss Fndng he prncpal componens from N observaons of X creaes an n n covarance marx Σ = XX. When N f n, hs s a convenen form of he covarance marx o use. An N N covarance marx resuls from X X and s useful hen n f N. Ths s ypcally he case hen an mage forms an observaon and n s very large. If / he SVD s used o decompose X as, X = UΛ V he n n covarance marx s found by: Σ = UΛ / V VΛ / U = UΛU () Ths can be recognzed as an egen-decomposon on XX here U s an n n marx hose columns are he egenvecors of XX, V s an N N marx hose columns are he egenvecors of X X and Λ s an n N marx hose frs r dagonal elemens correspond o non-zero egenvalues of he covarance marx n descendng order. Thus he r dmensonal subspace s formed by selecng he frs r ros of he ransformed daa marx X LD : X = U X (3) LD The N N covarance marx X X gves: X X = VΛ / U UΛ / V = VΛV (4) and he follong relaon may be used for dmensonaly reducon hen n f N (Foruna and Capson, 004): X LD = XV (5). Independen Componen Analyss ICA s orgnally developed for blnd source separaon hose goal s o recover muually ndependen bu unknon source sgnals from her lnear mxure hou knong he mxng coeffcens. ICA decorrelaes X by fndng a marx A such ha s s no jus decorrelaed bu sascally ndependen. The degree of ndependence s measured by he muual nformaon beeen he componens of he random varables : p( s) I ( s) = p( s)log ds (6) p ( ) k k sk here p (s) s he jon probably of s and p k ( s k ) are he margnal denses. If a nonlnear mappng y = g(s) s appled such ha y has unform margnal denses, has been shon ha muual nformaon s obaned by (Barle and Sejnosk, 997): I( y) = H ( y) = p( y) log p( y) dy (7) I (y) can hen be mnmzed h: [ ( A ] x) x I log dea = + E logg( s) = ( A) E g Aj Aj k A + (8) j here E [ ] denoes expeced value. Mulplyng by A A leads o he naural graden algorhm (Sh e al., 004): A ( I + E g( A x) x ) (9) [ ] A.3 Lnear Dscrmnan Analyss LDA crera are manly based on a famly of funcons of scaer marces. For example, he maxmzaon of r( Σ Σb ) or Σ / Σ s used, here b Σ, Σ are hn and beeen-class scaer marces, b respecvely. In LDA, he opmum lnear ransform s composed of r( n) egenvecors of Σ Σb correspondng o s r larges egenvalues. Alernavely, Σ Σ m can be used for LDA, here Σ m

3 represens he mxure scaer marx ( Σ = Σ + Σ ). m b A smple analyss shos ha boh Σ Σb and Σ Σm has he same egenvecor marxφ. In general, Σ b s no full rank, hence Σ m s used n place of Σ b. The compuaon of he egenvecor marxφ from Σ Σm s equvalen o he soluon of he generalzed egenvalue problem Σ mφ = Σ φλ, here Λ s he egenvalue marx (Fukunaga, 990). 3 SUPPORT VECTOR MACHINES To perform classfcaon h a lnear SVM, a labelled se of feaures { x, y} s consruced for all r feaures n he ranng daa se. The class of feaure c s defned by y = {, }. If he daa are assumed o be lnearly separable, he SVM aemps o fnd a separang hyperplane h he larges margn. The margn s defned as he shores dsance from he separang hyperplane o he closes daa pon. If he ranng daa follo: y ( x + b) 0 (0) Then he pons for hch he above equaly holds le on he hyperplanes x + b = and x + b =. The margn can be shon o be (Crsann and Shae-Taylor, 000): Margn = () The SVM aemps o fnd he par of hyperplanes hch gve he maxmum margn by mnmzng subjec o consrans on. Reformulang he problem usng he Lagrangan, he expresson o opmze for a nonlnear SVM can be ren as: r r r L( α) = α αα j y y jk( x, x j ) () = = j= K ( x, x ) s a kernel funcon sasfyng Mercer s condons. An example kernel funcon s he Gaussan radal bass funcon: x x K ( x, x ) = exp (3) σ here σ s he sandard devaon of he kernel s exponenal funcon. The decson funcon of he SVM can be descrbed by: p f ( x) = sgn y α K( x, x ) + b (4) = For daa pons hch le closes o he opmal hyperplane he correspondng α are non-zero, and hese are called suppor vecors. All oher parameers α are zero. As such, any modfcaon of he daa pons hch are no suppor vecors ll have no effec on he soluon. Ths ndcaes ha he suppor vecors conan all he necessary nformaon o reconsruc he hyperplane. 3. General Subspace Classfcaon An SVM can be used o classfy subspace feaures (ncludng PCA, ICA and LDA exraced feaures) as descrbed belo: ) The Transformaon marx A s deermned ran usng he ranng daa se X, ) The ranng and es daa ses n he reduced dmenson subspace are deermned as follos: ran ran es es S = A X, S = A X ran ) Defne daa pars ( s, y ) and apply a suppor vecor classfer o classfys es. 3. Ierave Subspace Classfcaon In order o mprove he generalzaon of he classfer, an erave algorhm hch moves ouler feaure vecors oard her class mean and modfes he bass vecors S o f he ne feaures has been proposed (Foruna and Capson, 004). We used hs algorhm h all hree feaure exracon mehods as follos: ran ) fnd A from X. ) nalze: ran ran es es S = A X, S = A X ) nalze he suppor vecor coeffcen marx Γ o he deny marx. v) repea v) move he suppor vecors oard he mean by an amoun proporonal o he suppor vecor α by: ran ran ran mean S = S Γ( S S ) v) recalculae A by: + ran A = X S here + denoes pseudo-nverse. v) calculae: ran ran es es S = A X, S = A X ran v) defne daa pars ( s, y ) and apply a suppor es vecor classfer o classfy S. 3

4 x) unl margn change p EXPERIMENTAL RESULTS 4. Gaussan Mxure Daa An example of o classes, each comprsng a mxure of hree Gaussan random varables, s used o llusrae he relaonshp beeen PCA, ICA and LDA exraced feaures classfed by SVM. The mxure of Gaussan daa pons X = [x c x c ] are defned by: X c 3 ) (x - n ) T = exp{( x - µ n Σn µ }, n = Σ n 6 X = c n = 4 Σ n here:, exp{( x -, n ) T µ n) Σn(x - µ } = = = Σ = Σ = Σ =.5.9 µ µ µ ,.9, µ 5,,, µ 4 = = µ = Σ4 = Σ5 = Σ6 = ORGINAL (a) ICA (c) Fgure : Example mxure of Gaussan daa se: (a) orgnal daa, (b) prncpal componen coeffcens, (c) ndependen componens coeffcens, (d) Lnear dscrmnan coeffcens. 0 -,,,, PCA (b) LDA - 0 (d) Fgure (.a) llusraes he dsrbuon of he orgnal daa pons n o dmensonal space. Fgures (.b-d) llusrae he ransformed daa by PCA, ICA and LDA, respecvely. Table () shos he classfcaon resuls usng drec and erave mplemenaon of each mehod. As shon, LDA exraced feaures provde slghly mproved recognon performance compared o PCA and ICA feaures. Table : Classfcaon resuls for mxure of Gaussan daa se Mehod Mean of Margn Mean of SV Mean of Recognon Rae No subspace PCA ICA LDA PCA erave ICA erave LDA erave Facal Image Daabase The developed algorhms ere also appled o Yale face daabase B (Georghades, e al., 00). For hs expermen, class recognon expermens are performed over 36 pars of subjecs. For each par of subjecs, a ranng daa se s consruced from he frs 3 lghng posons for pose and of each subjec. The es daa se comprsed he same par of subjecs maged under he las 3 lghng poson from pose 7 and 8. The ranng and es mages ere hsogram equalzed and mean cenred before subspace calculaon and classfcaon. For hs example n f N, so e used X X o compue he egenvecors. Recognon performance (margn, number of suppor vecors and error rae) as esed for each subjec par for kernel σ rangng from o 5. The dmensonaly of he ranng subspace s reduced o 5 pror o recognon. Fgures (.a) and (.b) sho he ranng mages for o faces (seleced randomly) from he daa se. Fgs. (c) and (d) sho he es mages for he same o faces. Fg. 3 shos he resulng prncpal, ndependen and lnear bass mages for he ranng mages shon n Fgs. (a) and (b). Table () shos he average number of suppor vecors, margn and recognon rae for he enre daa se. As llusraed n Table (), erave algorhms provde beer generalzaon of he SVM classfer. The mprovemen n generalzaon he erave echnques llusraed by 4

5 mproved margn and reduced number of suppor vecors s sascally sgnfcan for all resuls on he face daabase. (a) (b) and he number of suppor vecors compared o ra daa and PCA componen represenaons. Table : Classfcaon resuls for Yale face Mehod Mean of Margn Mean of SV Mean of Recognon Rae No subspace PCA ICA LDA PCA erave ICA erave LDA erave CONCLUDING REMARKS (c) Fgure : Example of ranng and es mages: (a) class ranng, (b) class ranng, (c) class es, (d) class es. (d) (a) (b) (a) (b) (c) In hs paper, e used hree feaure exracon mehods ncludng PCA, ICA and LDA o reduce he dmensonaly of he ranng space. An erave algorhm as ulzed o furher enhance he generalzaon ably of he feaure exracon mehods by producng compac classes. Our expermenal resuls on smulaed daa llusraed ha he proposed mehods mprove he performance of he SVM classfer boh n erms of accuracy and complexy. These resuls also llusraed ha LDA provdes slghly mproved generalzaon compared o PCA and ICA. Expermenal resuls on a facal daabase demonsraed he same mprovemen n classfcaon performance usng LDA exraced feaures. In our fuure ork, e plan o evaluae he performance of adapve PCA and LDA algorhms for feaure exracon n facal daa. REFERENCES (d) (e) (f) Fgure 3: Example componens (conras enhanced): (a) mages of PCA, (b) mages of ICA, (c) mages of LDA, (d) mages of erave PCA, (e) mages of erave ICA, (f) mages of erave LDA. Moreover, among hree feaure exracon algorhms, LDA componen represenaon exhbs hgher performance h respec o margn, number of suppor vecors and recognon rae. In all of he expermens, ICA conssenly ncreased he margn Sun, Z., Bebs, G., Mller, R., 004. Objec deecon usng feaure subse selecon. Paern Recognon, Elsever Vol. 37, No., pp Jan, A., Dun, R., Mao, J., 00. Sascal paern recognon: a reve. IEEE Trans. Paern Anal. Machne Inell, Vol., No., pp Belhumeur, P. N., Hespanha, J. P., Kregman, D. J., 997. Egenfaces vs. Fsherfaces: Recognon usng class specfc lnear projecon. IEEE Trans. Paern Anal. Machne Inell, Vol. 9, pp Moghaddam, B., 00. Prncpal manfolds and probablsc subspaces for vsual recognon. IEEE Trans. Paern Anal. Machne Inell, Vol. 4, No. 6, pp Ohman, H., Aboulnasr, T., 003. A separable lo complexy D HMM h applcaon o face recognon. 5

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