Using Mixture Covariance Matrices to Improve Face and Facial Expression Recognitions

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1 Usng Mxture Covarance Matrces to Improve Face and Facal Expresson Recogntons Carlos E. homaz, Duncan F. Glles and Raul Q. Fetosa 2 Imperal College of Scence echnology and Medcne, Department of Computng, 80 Queen s Gate, London SW7 2BZ, Unted Kngdom {cet, 2 Catholc Unversty of Ro de Janero, Department of Electrcal Engneerng, r. Marques de Sao Vcente 225, Ro de Janero , Brazl Abstract. In several pattern recognton problems, partcularly n mage recognton ones, there are often a large number of features avalable, but the number of tranng samples for each pattern s sgnfcantly less than the dmenson of the feature space. hs statement mples that the sample group covarance matrces often used n the Gaussan maxmum probablty classfer are sngular. A common soluton to ths problem s to assume that all groups have equal covarance matrces and to use as ther estmates the pooled covarance matrx calculated from the whole tranng set. hs paper uses an alternatve estmate for the sample group covarance matrces, here called the mxture covarance, gven by an approprate lnear combnaton of the sample group and pooled covarance matrces. Experments were carred out to evaluate the performance assocated wth ths estmate n two recognton applcatons: face and facal expresson. he average recognton rates obtaned by usng the mxture covarance matrces were hgher than the usual estmates. Introducton A crtcal ssue for the Gaussan maxmum probablty classfer s the nverse of the sample group covarance matrces. Snce n practce these matrces are not nown, estmates must be computed based on the observatons (patterns) avalable n a tranng set. In some applcatons, however, there are often a large number of features avalable, but the number of tranng samples for each group s lmted and sgnfcantly less than the dmenson of the feature space. hs mples that the sample group covarance matrces wll be sngular. hs problem, whch s called a small sample sze problem [5], s qute common n pattern recognton, partcularly n mage recognton where the number of features s very large. One way to overcome ths problem s to assume that all groups have equal covarance matrces and to use as ther estmates the weghtng average of each sample group covarance matrx, gven by the pooled covarance matrx calculated from the whole tranng set. hs paper uses another estmate for the sample group covarance matrces [4], here called mxture covarance matrces, gven by an approprate lnear combnaton of the sample group covarance matrx and the pooled covarance one. he mxture covarance matrces have the property of havng the same ran as the pooled estmate, whle allowng

2 a dfferent estmate for each group. hus, the mxture estmate may result n hgher accuracy. In order to evaluate ths approach, two pattern recognton applcatons were consdered: face recognton and facal expresson recognton. he evaluaton used dfferent mage databases for each applcaton and a probablstc model combnes the well-nown dmensonalty reducton technque called Prncpal Component Analyss (PCA) and the Gaussan maxmum probablty classfer to nvestgate the mxture covarance matrces on the referred recognton tass. Experments carred out show that the mxture covarance estmates attaned the best performance n both applcatons consdered. 2 Dmensonalty Reducton One of the most successful approaches to the problem of creatng a low dmensonal mage representaton s based on Prncpal Component Analyss (PCA). PCA generates a set of orthonomal bass vectors, nown as prncpal components, that mnmzes the mean square reconstructon error and descrbe major varatons n the whole tranng set consdered. he reasonng behnd applyng PCA frst for dmensonalty reducton nstead of analyzng the maxmum probablty classfer drectly on the face or facal expresson mages s based on the orgnal hgh-dmensonal space. As the number of tranng samples s lmted and sgnfcantly less than the number of pxels of each mage, the hghdmensonal space s ndeed sparsely represented, mang the parameter estmaton qute complcated ths behavour s called the curse of dmensonalty [8]. Furthermore, many researchers have confrmed that the PCA representaton has good generalzaton ablty especally when the dstrbutons of each class are separated by the mean dfference [,6,7,9]. 3 Maxmum Probablty Classfer he basc problem n the decson-theoretc methods for pattern recognton conssts of fndng a set of g dscrmnant functons d (x), d 2 (x),..., d g (x), where g s the number of groups or classes, wth the decson rule such that f the p-dmensonal pattern vector x belongs to the class ( g), then d (x) d j (x), for all j and j g. he Bayes classfer desgned to maxmze the total probablty of correct classfcaton, where equal pror probabltes for all groups are assumed, corresponds to a set of dscrmnant functons equal to the correspondng probablty densty functons, that s, d (x)=f (x) for all classes [8]. he most common probablty densty functon appled to pattern recognton systems s based on the Gaussan multvarate dstrbuton d ( x) = f ( x µ, Σ ) = exp ( x µ ) ( ) p / 2 / 2 Σ x µ, (2π) Σ 2 () where µ and Σ are the class populaton mean vector and covarance matrx. Usually the true values of the mean and the covarance matrx are seldom nown and must be estmated from tranng samples. he mean s estmated by the usual sample mean

3 x, x = j = x µ, (2) where j s observaton j from class, and s the number of tranng observatons from class. he covarance matrx s commonly estmated by the sample group covarance matrx defned as Σ S = ( ) j=, j ( x, x )( x, x ) j j. (3) From replacng the true values of the mean and the covarance matrx n () by ther respectve estmates, the Bayes decson rule acheves optmal classfcaton accuracy only when the number of tranng samples ncreases toward nfnty [4]. In fact for p- dmensonal patterns the sample covarance matrx s sngular f less than p + tranng samples from each class are avalable, that s, the sample covarance matrx can not be calculated f s less than the dmenson of the feature space. One method routnely appled to solve ths problem s to assume that all classes have equal covarance matrces, and to use as ther estmates the pooled covarance matrx. hs covarance matrx s a weghtng average of each sample group covarance matrx and, assumng that all classes have the same number of tranng observatons, s gven by S = g pooled S g =. (4) Snce more observatons are taen to calculate the pooled covarance matrx S pooled, ths one wll potentally have a hgher ran than S and wll be eventually full ran. Although the pooled estmate does provde a soluton for the algebrac problem arsng from the nsuffcent number of tranng samples n each group, assumng equal covarance for all groups may brng about dstortons n the modelng of the classfcaton problem and consequently lower accuracy. 4 Mxture Covarance Matrx he choce between the sample group covarance matrx and the pooled covarance one represents a restrctve set of estmates for the true covarance matrx. A less lmted set can be obtaned usng the mxture covarance matrx. 4. Defnton he mxture covarance matrx s a lnear combnaton between the pooled covarance matrx S pooled and the sample covarance matrx of each class S. It s gven by Smx ( w ) w S ( w ) S = pooled +. (5)

4 he mxture parameter w taes on values 0 < w and s dfferent for each class. hs parameter controls the degree of shrnage of the sample group covarance matrx estmates toward the pooled one. Each Smx matrx has the mportant property of admttng an nverse f the pooled estmate S pooled does so [2]. hs mples that f the pooled estmate s non-sngular and the mxture parameter taes on values w > 0, then the Smx wll be non-sngular. hen the remanng queston s: what s the value of the w that gves a relevant lnear mxture between the pooled and sample covarance estmates? A method that determnes an approprate value of the mxture parameter s descrbed n the next secton. 4.2 he mxture parameter Accordng to Hoffbec and Landgrebe [4], the value of the mxture parameter w can be approprately selected so that a best ft to the tranng samples s acheved. her technque s based on the leave-one-out-lelhood (L) parameter estmaton. In the L method, one sample of the class tranng set s removed and the mean and covarance matrx from the remanng samples are estmated. hen the lelhood of the excluded sample s calculated gven the prevous mean and covarance matrx estmates. hs operaton s repeated tmes and the average log lelhood s computed over all the samples. her strategy s to evaluate several dfferent values of w n the range 0 < w, and then choose w that maxmzes the average log lelhood. he mean of class wthout sample r may be computed as x \ r = x, j x, r. (6) ( ) j = he notaton \r ndcates the correspondng quantty s calculated wth the r th observaton from class removed. Followng the same dea, the sample covarance matrx and the pooled covarance matrx of class wthout sample r are S\ r = ( x, j x \ r )( x, j x \ r ) ( x, r x \ r )( x, r x \ r ) ( 2) j =, (7) g S pooled = S j S + S r r \. (8) \ g j = hen the average log lelhood of the excluded samples can be wrtten as follows: L ( w ) = ln[ f ( x, r x \ r, Smx \ r ( w ))], (9) r = where ( x x, Smx ( w )) r f s the Gaussan probablty functon defned n () wth, r \ r \ r x \ mean vector and Smx ( w ) covarance matrx defned as Smx \ r = +. (0) \ r ( w ) w S pooled ( w ) S \ r \ r

5 hs approach, f mplemented n a straghtforward way, would requre computng the nverse and determnant of the Smx\ r ( w ) for each tranng sample. As the Smx\ r ( w ) s a p by p matrx and p s typcally a large number, ths computaton would be qute expensve. Hoffbec and Landgrebe [4], usng the Sherman-Morrson-Woodbury formula [3], have showed that t s possble to sgnfcantly reduce the requred computaton by wrtng the mxture covarance matrx n a form as follows: ln where d [ ( )] = [ ] f x x, Smx ( w ) ln Q ( vd), r \ r \ r, () 2 2 vd ( ) Q = ( w ) + w S + w S pooled, (2) ( 2) g( 2) 2 v = ( )( w 2) ( g ), g (3) ( x, r x ) Q ( x, r d = x ). (4) Once the mxture parameter w that maxmzes the average log lelhood s selected, the proposed covarance matrx estmate s calculated usng all the tranng samples and replaced nto the maxmum probablty classfer defned n (). 5 Experments wo experments wth two dfferent databases were performed. In the face recognton experment the ORL Face Database contanng ten mages for each of 40 ndvduals, a total of 400 mages, were used. he ohou Unversty has provded the database for the facal expresson experment. hs database s composed of 93 mages of expressons posed by nne Japanese females. Each person posed three or four examples of each sx fundamental facal expresson: anger, dsgust, fear, happness, sadness and surprse. he database has at least 29 mages for each fundamental facal expresson. For mplementaton convenence all mages were frst reszed to 64x64 pxels. he experments were carred out as follows. Frst PCA reduces the dmensonalty of the orgnal mages and secondly the Gaussan maxmum probablty classfer usng one out of the three covarance estmates (S, S pooled and Smx ) was appled. Each experment was repeated 25 tmes usng several PCA dmensons. Dstnct tranng and testng sets were randomly drawn, and the mean and standard devaton of the recognton rate were calculated. he face recognton classfcaton was computed usng for each ndvdual 5 mages to tran and 5 mages to test. In the facal expresson recognton, the tranng and test sets were respectvely composed of 20 and 9 mages. he sze of the mxture parameter (0 < w ) optmzaton range was taen to be 20, that s w = [0.05, 0.0, 0.5,, ].

6 6 Results ables and 2 present the tranng and test average recognton rates (wth standard devatons) of the face and facal expresson databases, respectvely, over the dfferent PCA dmensons. Snce only 5 mages of each ndvdual were used to form the face recognton tranng set, the results relatve to the sample group covarance estmate were lmted to 4 PCA components. able shows that n all but one experment the Smx estmate led to hgher accuracy than dd both the pooled covarance and sample group covarance matrces. In terms of how senstve the mxture covarance results were to the choce of the tranng and test sets, t s far to say that the Smx standard devatons were smlar to the other two covarance estmates. able 2 shows the results of the facal expresson recognton. For more than 20 components when the sample group covarance estmate became sngular, the mxture covarance estmate reached hgher recognton rates than the pooled covarance estmate. Agan, regardng the computed standard devatons, the Smx estmate showed to be as senstve to the choce of the tranng and test sets as the other two estmates. PCA Sgroup Spooled Smx Components ranng est ranng est ranng est (0.4) 5.6 (4.4) 73.3 (3.) 59.5 (3.0) 90. (2.) 70.8 (3.2) (.2) 88.4 (.4) 99.4 (0.5) 92.0 (.5) (0.6) 9.8 (.8) 00.0 (0.) 94.5 (.7) (0.2) 94.7 (.7) 00.0 (0.0) 95.9 (.5) (0.0) 95.4 (.5) 00.0 (0.0) 96.2 (.6) (0.0) 95.7 (.2) 00.0 (0.0) 96.4 (.5) (0.0) 95.0 (.6) 00.0 (0.0) 95.8 (.6) (0.0) 94.9 (.6) 00.0 (0.0) 95.4 (.6) able. Face Recognton Results PCA Sgroup Spooled Smx Components ranng est ranng est ranng est (4.2) 20.6 (3.9) 32.3 (3.0) 2.6 (3.8) 34.9 (3.3) 2.3 (4.) (3.6) 38.8 (5.6) 49.6 (3.9) 26.5 (6.8) 58.5 (3.7) 27.9 (5.6) (0.5) 64.3 (6.4) 69. (3.6) 44.4 (5.3) 82.9 (2.9) 49.7 (7.7) (2.6) 55.9 (7.7) 9.4 (2.8) 6.3 (7.) (2.8) 64.9 (6.9) 94.8 (2.2) 68.3 (5.) (.7) 70. (7.8) 96.8 (.3) 72.3 (6.2) (.7) 72.0 (7.4) 97.7 (.) 75.6 (5.5) (.4) 75.6 (7.) 98.3 (.) 77.2 (5.7) (.3) 78.4 (6.5) 98.6 (0.8) 79. (5.4) (.0) 79.4 (5.8) 99.2 (0.7) 8.0 (6.6) (0.9) 8.6 (6.6) 99.5 (0.6) 82.8 (6.3) (0.8) 82. (5.9) 99.6 (0.6) 83.6 (7.2) (0.6) 83.3 (5.5) 99.8 (0.4) 84.5 (6.2) able 2. Facal Expresson Recognton Results

7 7 Concluson hs paper used an estmate for the sample group covarance matrces, here called mxture covarance matrces, gven by an approprate lnear combnaton of the sample group covarance matrx and the pooled covarance one. he mxture covarance matrces have the property of havng the same ran as the pooled estmate, whle allowng a dfferent estmate for each group. Extensve experments were carred out to evaluate ths approach on two recognton tass: face recognton and facal expresson recognton. A Gaussan maxmum probablty classfer was bult usng the mxture estmate and the typcal sample group and pooled estmates. In both tass the mxture covarance estmate acheved the hghest accuracy. Regardng the senstveness to the choce of the tranng and test sets, the mxture covarance matrces presented smlar performance to the other two usual estmates. References. C. Lu and H. Wechsler, Learnng the Face Space Representaton and Recognton. In Proc. 5 th Int l Conference on Pattern Recognton, ICPR 2000, Barcelona, Span, September C.E. homaz, R.Q. Fetosa, A. Vega, Separate-Group Covarance Estmaton wth Insuffcent Data for Object Recognton. In Proc. Ffth All-Uranan Internatonal Conference, pp. 2-24, Urane, G.H. Golub and C.F. Van Loan, Matrx Computatons, second edton. Baltmore: Johns Hopns Unv. Press, J.P. Hoffbec and D.A. Landgrebe, "Covarance Matrx Estmaton and Classfcaton wth Lmted ranng Data", IEEE rans. Pattern Analyss and Machne Intellgence, vol. 8, no. 7, July K. Fuunaga, Introducton to Statstcal Pattern Recognton, second edton. Boston: Academc Press, M. Krby and L. Srovch, "Applcaton of the Karhunen-Loeve procedure for the characterzaton of human faces", IEEE rans. Pattern Analyss and Machne Intellgence, Vol. 2, No., Jan M. ur and A. Pentland, Egenfaces for Recognton, Journal of Cogntve Neuroscence, Vol. 3, pp , R.A. Johnson R.A. and D.W. Wchern, Appled Multvarate Statstcal Analyss, by Prentce- Hall, Inc., 3d. edton, W. Zhao, R. Chellappa and A. Krshnaswamy, Dscrmnant Analyss of Prncpal Components for Face Recognton, Proc. 2nd Internatonal Conference on Automatc Face and Gesture Recognton, pp , Japan, Aprl, 998.

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