Small Sample Problem in Bayes Plug-in Classifier for Image Recognition

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1 mall amle Problem n Bayes Plug-n Classfer for Image Recognton Carlos E. homaz, Duncan F. Glles and Raul Q. Fetosa Imeral College of cence echnology and Medcne, Deartment of Comutng, 80 Queen s Gate, London W7 BZ, Unted Kngdom {cet,dfg}@doc.c.ac.u Catholc Unversty of Ro de Janero, Deartment of Electrcal Engneerng, r. Marques de ao Vcente 5, Ro de Janero , Brazl raul@ele.uc-ro.br Abstract. Image attern recognton roblems, esecally face and facal exresson ones, are commonly related to small samle sze roblems. In such alcatons there are a large number of features avalable but the number of tranng samles for each attern s consderably less than the dmenson of the feature sace. he Bayes lug-n classfer has been successfully aled to dscrmnate hgh dmensonal data. hs classfer s based on smlarty measures that nvolve the nverse of the samle grou covarance matrces. hese matrces, however, are sngular n small samle sze roblems. herefore, other methods of covarance estmaton have been roosed where the samle grou covarance estmate s relaced by covarance matrces of varous forms. In ths aer, two Bayes lug-n covarance estmators nown as RDA and LOOC are descrbed and a new covarance estmator s roosed. he new estmator does not requre an otmsaton rocedure, but an egenvector-egenvalue orderng rocess to select nformaton from the rojected samle grou covarance matrces whenever ossble and the ooled covarance otherwse. he effectveness of the method s shown by exermental results carred out on face and facal exresson recognton, usng dfferent databases for each alcaton. Introducton Face and facal exresson recognton are examles of attern recognton roblems where there are a large number of features avalable, but the number of tranng samles for each grou or class s lmted and sgnfcantly less than the dmenson of the feature sace. hs roblem, whch s called a small samle sze roblem [Fu90], s ndeed qute common nowadays, esecally n mage recognton alcatons where atterns are usually comosed of a huge number of xels. he Bayes lug-n classfer has been successfully aled to dscrmnate hgh dmensonal data [HL96, CW98, L99, GF0]. hs classfer s based on smlarty measures that nvolve the nverse of the true covarance matrx of each class. nce n ractcal

2 cases these matrces are not nown, estmates must be comuted based on the atterns avalable n a tranng set. he usual choce for estmatng the true covarance matrces s the maxmum lelhood estmator defned by the corresondng samle grou covarance matrces. However, n small samle sze alcatons the samle grou covarance matrces are sngular. One way to overcome ths roblem s to assume that all grous have equal covarance matrces and to use as ther estmates the weghtng average of each samle grou covarance matrx, gven by the ooled covarance matrx calculated from the whole tranng set. he decson concernng whether to choose the samle grou covarance matrces or the ooled covarance matrx reresents a lmted set of estmates for the true covarance matrces [Fr89]. herefore, other aroaches have been aled not only to overcome the small sze effect but also to rovde hgher classfcaton accuracy. In ths aer, two Bayes lug-n covarance estmators nown as RDA [Fr89] and LOOC [HL96] are descrbed and a new covarance estmator s roosed. Exerments were carred out to evaluate and comare these aroaches on face and facal exresson recognton, usng dfferent databases for each alcaton. he effectveness of the new covarance method s shown by the results. he Bayes Plug-n Classfer he Bayes lug-n classfer, also called the Gaussan maxmum lelhood classfer, s based on the -multvarate normal or Gaussan class-condtonal robablty denstes ( x π ) f ( x µ, Σ ) = ex ( x µ ) Σ ( ) / / x µ, (π ) Σ () where µ and Σ are the true class oulaton mean vector and covarance matrx. he notaton denotes the determnant of a matrx. he class-condtonal robablty denstes f x µ, Σ ) defned n () are also nown ( as the lelhood densty functons. Assumng that all of the g grous or classes have the

3 same ror robabltes, the Bayes classfcaton rule can be secfed as: Assgn attern x to class f f ( x µ, Σ ) = max f j ( x µ j, Σ j ) () j g Another way to secfy equaton () s to tae the natural logarthms of the quanttes nvolved. hus, the otmal Bayes classfcaton rule defned n equaton () may be smlfed to: Assgn attern x to class f d ( x) = mn[ln Σ j + ( x µ j ) Σ j ( x µ j )] = mn d j ( x) (3) j g j g he Bayes classfcaton descrbed n (3) s also nown as the quadratc dscrmnant rule (QD). In ractce, however, the true values of the mean and covarance matrx,.e. µ and Σ, are seldom nown and must be relaced by ther resectve estmates calculated from the tranng samles avalable. he mean s estmated by the usual samle mean x whch s the maxmum lelhood estmator of µ. he covarance matrx s commonly estmated by the samle grou covarance matrx whch s the unbased maxmum lelhood estmator of Σ. From relacng the true values of the mean and covarance matrx n (3) by ther resectve estmates ( lug-n ), the QD rule can be rewrtten as: Assgn attern x to class that mnmzes the generalzed dstance between x and x d ( x) = ln + ( x x ) ( x x ). (4). mall amle ze Problems Although x and are maxmum lelhood estmators of µ and Σ, t s nown that the msclassfcaton rate defned n (4) aroaches the otmal rate obtaned by equaton (3) only when the samle szes n the tranng set aroach nfnty [And84]. Furthermore, the erformance of (4) can be serously degraded n small samles due to the nstablty of x and estmators. In fact, for -dmensonal atterns the use of s esecally roblematc f less than + tranng observatons from each class are avalable, that s, the 3

4 samle grou covarance matrx s sngular f the number of observatons of each grou s less than the dmenson of the feature sace. One method routnely aled to overcome the small samle sze roblem and consequently deal wth the sngularty and nstablty of the dscrmnant rule (LD) whch s obtaned by relacng the n (4) wth the ooled samle covarance matrx s to emloy the so-called lnear = n ) + ( n ) ( N g + + ( n g ) g, (5) where n s the number of tranng observatons from class, g s the number of grous or classes and N n + n + + n. nce more observatons are taen to calculate the = g ooled covarance matrx, s ndeed a weghted average of the, wll otentally have a hgher ran than (and would be eventually full ran) and the varances of ther elements are smaller than the varances of the corresondng elements. heoretcally, however, s a consstent estmator of the true covarance matrces Σ only when Σ =Σ = =Σ g. 3 Covarance Estmators he decson concernng whether to choose the samle grou covarance matrces (or QD classfer) or the ooled covarance matrx (or LD classfer) reresents a lmted set of estmates for the true covarance matrces Σ, artcularly n small samle sze roblems. herefore, other aroaches have been aled not only to overcome these roblems but also to rovde hgher classfcaton accuracy. In the next secton, two mortant aroaches are descrbed and a new covarance estmator s roosed. 3. Dscrmnant Reguralzaton Method Reguralzaton methods have been successfully used n solvng small samle sze roblems [Osu86]. In fact, Fredman [Fr89] has roosed one of the most mortant regularzaton rocedures called reguralzed dscrmnant analyss (RDA). 4

5 he Fredman s aroach s bascally a two-dmensonal otmsaton method that shrns both the towards and also the egenvalues of the towards equalty by blendng the frst shrnage wth multles of the dentty matrx. In ths context, the samle covarance matrces of the dscrmnant rule defned n (4) are relaced by the followng rda (, γ ) rda rda rda rda tr γ γ γ ( ( )) (, ) = ( ) ( ) + I, ( )( n ) + ( N g) ( ) = ( ) n + N, (6) where the notaton tr denotes the trace of a matrx. hus the regularzaton arameter controls the degree of shrnage of the samle grou covarance matrx estmates toward the ooled covarance one, whle the arameter γ controls the shrnage toward a multle of the dentty matrx. nce the multler rda rda / tr( ( )) s just the average egenvalue of (), the shrnage arameter γ has the effect of decreasng the larger egenvalues and ncreasng the smaller ones [Fr89]. he RDA mxng arameters and γ are restrcted to the range 0 to (otmsaton grd) and are selected to maxmse the leave-one-out classfcaton accuracy based on the corresondng rule defned n (4). In other words, the followng classfcaton rule s develoed on the N tranng observatons exclusve of a artcular observaton x, v and then used to classfy x, v : Choose class such that rda d ( x, v j, v j g ) = mn d ( x ), wth rda ( (, γ )) ( x x ) d j ( x, v ) = ln j / v (, γ ) + ( x, v x j / v ) j / v, v j / v. where the notaton /v reresents the corresondng quantty wth observaton (7) x, v removed. Each of the tranng observatons s n turn held out and then classfed n ths manner. he resultng msclassfcaton loss,.e. the number of cases n whch the observaton left out s allocated to the wrong class averaged over all the tranng observatons, s then used to choose the best grd-ar (, γ ). 5

6 Although Fredman s RDA method s theoretcally a well-establshed aroach, t has ractcal drawbacs. Deste the substantal amount of comutaton saved by tang advantage of matrx udatng formulas [Fr89], RDA s a comutatonally ntensve method. For each ont on the two-dmensonal otmsaton grd, RDA requres the evaluaton of the roosed estmates of every class. In stuatons where a large number of g grous s consdered, the RDA seems to be unfeasble. In addton, as RDA maxmses the classfcaton accuracy calculatng all covarance estmates smultaneously, t s restrcted to usng the same value of the mxng arameters for all the classes. hese same values may not be otmal for all classes. 3. Leave-One-Out Lelhood Method In ractcal stuatons, t seems arorate to allow covarance matrces to be estmated by dstnct mxng arameters. Hoffbec [HL96] has roosed a leave-one-out covarance estmator (LOOC) that deends only on covarance estmates of sngle classes. In LOOC each covarance estmate s otmsed ndeendently and a searate mxng arameter s comuted for each class based on the corresondng lelhood nformaton. he dea s to examne ar-wse mxtures of the samle grou covarance estmates and the unweghted common covarance estmate, defned as = g g =, (8) together wth ther dagonal forms. he LOOC estmator has the followng form: looc ( α )dag( ) + α 0 α ( α ) = ( α ) + ( α ) < α. (9) (3 α ) + ( α )dag( ) < α 3 he mxng or shrnage arameter α determnes whch covarance estmate or mxture of covarance estmates s selected, that s: f α = 0 then the dagonal of samle covarance s used; f α = the samle covarance s returned; f α = the common covar- ance s selected; and f α = 3 the dagonal form of the common covarance s consdered. Other values of α lead to mxtures of two of the aforementoned estmates [HL96]. 6

7 he Hoffbec strategy s to evaluate several values of α over the otmsaton grd 0 α 3, and then choose α that maxmzes the average log lelhood of the corresondng -varate normal densty functon, comuted as follows: n looc ( ( )) looc LOOL ( α ) = ln / v ( α ) ( x, v x / v ) / v α ( x, v x / v ) n, (0) v= where the notaton /v reresents the corresondng quantty wth observaton x, v left out. Once the mxture arameter α s selected, the corresondng leave-one-out covarance looc estmate ( α ) s calculated usng all the n tranng observatons and substtuted for nto the Bayes dscrmnant rule defned n (4) [HL96]. As can be seen, the comutaton of the LOOC estmate requres only one densty functon be evaluated for each ont on the α one-dmensonal otmsaton grd, but also looc ( nvolves calculatng the nverse and determnant of the ( by ) matrx α ) for each tranng observaton belongng to the th class. Analogously, Hoffbec has reduced the LOOC requred comutaton by consderng vald aroxmatons of the covarance estmates and usng ran-one udatng formulas [HL96]. herefore, the fnal form of LOOC requres less comutaton than RDA estmator. 4 A New Covarance Estmator Accordng to Fredman s RDA and Hoffbec s LOOC aroaches descrbed n the revous secton, and several other smlar methods [GR89, GR9, L99] not descrbed n ths reort, otmsed lnear combnatons of the samle grou covarance matrces and, for nstance, the ooled covarance matrx not only overcome the small samle sze roblem but also acheve better classfcaton accuracy than LD and standard QD classfers. However, n stuatons where are sngular, such aroaches may lead to nconvenent basng mxtures. hs statement, whch s better exlaned n the followng subsecton, forms the bass of the new covarance estmator dea, called Covarance Projecton Orderng method. 7

8 4. Covarance Projecton Orderng Method he Covarance Projecton Orderng (COPO) estmator examnes the combnaton of the samle grou covarance matrces and the ooled covarance matrx n the QD classfers usng ther sectral decomoston reresentatons. hs new estmator has the roerty of havng the same ran as the ooled estmate, whle allowng a dfferent estmate for each grou. Frst, n order to understand the aforementoned nconvenent basng mxtures, let a matrx mx be gven by the followng lnear combnaton: mx = a + b, () where the mxng arameters a and b are ostve constants, and the ooled covarance matrx the matrces s a non-sngular matrx. he mx Φ and mx Λ formula [Fu90], t s ossble to wrte mx egenvectors and egenvalues are gven by, resectvely. From the covarance sectral decomoston mx 0 mx mx mx mx mx mx mx mx Φ ) Φ = Λ = = dag[,,..., ], () mx 0 ( where mx mx, mx,..., are the mx egenvalues and s the dmenson of the measurement sace consdered. Usng the nformaton rovded by equaton (), equaton () can be rewrtten as: ( Φ mx ) mx Φ mx = dag[ = ( Φ = a( Φ = aλ mx mx, mx ) mx mx ) Φ * * + bλ [ a + b ] * = dag[ a + b,..., * mx ] mx Φ mx ) + b( Φ *, a + b * Φ mx *,..., a + b * ] (3) * * * * * * where,,..., and,,..., are the corresondng sread values of samle grou covarance and ooled covarance matrces sanned by the mx egenvectors matrx mx Φ becomes. hen, the dscrmnant score of the QD rule n sectral decomoston form 8

9 mx d ( x) = * ( a + b ) + mx [( φ ) ( ln * = = a + * x x )], * (4) b where φ s the corresondng -th egenvector of the matrx mx. As can be observed, the dscrmnant score descrbed n equaton (4) consders the dsersons of samle grou covarance matrces sanned by all the herefore, n roblems where the grou samle szes mx egenvectors. n are small comared wth the dmenson of the feature sace, the corresondng ( n + ) dserson values are often estmated to be 0 or aroxmately 0 and the dscrmnant score seems to be not convenent. In fact, a lnear combnaton of the samle grou covarance matrx and the ooled covarance n a subsace where the former s oorly reresented may dstort the otmsaton rocedure because t uses the same arameters for the whole sace. Other covarance estmators have not addressed ths roblem. he COPO estmator s a smle aroach to overcome ths roblem. Bascally, the dea s to use all the samle grou covarance nformaton avalable whenever ossble and the ooled covarance nformaton otherwse. Regardng equatons () and (3), ths dea can be derved as follows: coo = + = = coo = * * f ran( ), otherwse. coo coo φ ( φ coo ), where a = b = (5) hen the dscrmnant scored descrbed n equaton (4) becomes: d ( x) = r = * * [( φ ) ( x x )] [( φ ) ( x x )] ln + ln + + *, * (6) = r+ where r = ran ) and ordered n he COPO estmator rovdes a new combnaton of the samle grou covarance matrces ( coo φ r = * decreasng values. coo = r+ coo s the corresondng -th egenvector of the matrx coo and the ooled covarance matrx strongly related to the ran of n such a way that ths combnaton s or, equvalently, to the number of tranng samles n. It can be vewed as a -dmensonal non-sngular aroxmaton of an r-dmensonal sn- 9

10 gular matrx. he COPO method does not requre an otmsaton rocedure, but an egenvector-egenvalue orderng rocess to select nformaton from the rojected samle grou covarance matrces whenever ossble and the ooled covarance otherwse. herefore, the comutatonal ssues regardng the Covarance Projecton Orderng aroach s less severe than the Fredman s RDA and Hoffbec s LOOC aroaches. In addton, the COPO method s not restrcted to use the same covarance combnaton for all classes, allowng covarance matrces to be dstnctly estmated. 5 Exerments and Results In order to evaluate the Covarance Projecton Orderng (COPO) aroach, two mage recognton alcatons were consdered: face recognton and facal exresson recognton. In the face recognton exerments the ORL Face Database was used, whch contans ten mages for each of 40 ndvduals, a total of 400 mages. he ohou Unversty has rovded the database for the facal exresson exerment. hs database [LBA99] s comosed of 93 mages of exressons osed by nne Jaanese females. Each erson osed three or four examles of each sx fundamental facal exresson: anger, dsgust, fear, haness, sadness and surrse. he database has at least 9 mages for each fundamental facal exresson. For mlementaton convenence all mages were frst reszed to 64x64 xels. 5. Exerments he exerments were carred out as follows. Frst the well-nown dmensonalty reducton technque called Prncal Comonent Analyss (PCA) [P9] reduced the dmensonalty of the orgnal mages and secondly the Bayes lug-n classfer usng one of the fve covarance estmators was aled: ) amle grou covarance grou; ) Pooled covarance ooled; 3) Covarance rojecton orderng coo; 4) Fredman s covarance rda; 5) Hoffbec s covarance looc. he ORL database s avalable free of charge, see htt:// 0

11 Each exerment was reeated 5 tmes usng several PCA dmensons. Dstnct tranng and test sets were randomly drawn, and the mean and the standard devaton of the recognton rate were calculated. he face recognton classfcaton was comuted usng for each ndvdual n the ORL database 5 mages to tran and 5 mages to test. In the facal exresson recognton, the tranng and test sets were resectvely comosed of 0 and 9 mages. he RDA otmsaton grd was taen to be the outer roduct of = [0,0.5,0.354,0.650,.0] and γ = [0,0.5,0.5,0.75,.0 ], dentcally to the Fredman s wor [Fr89]. Analogously, the sze of the LOOC mxture arameter [HL96] was α = [0,0.5,0.5,...,.75,3.0]. 5. Results ables and resent the tranng and test average recognton rates (wth standard devatons) of the ORL and ohou face and facal exresson databases, resectvely, over the dfferent PCA dmensons. Also resented are the mean of the otmsed RDA and LOOC arameters. For the ORL face database, only 6 LOOC arameters corresondng to the subjects, 5, 0, 0, 30 and 40 are shown. he notaton - n the grou rows ndcate that the samle grou covarance were sngular and could not classfy the samles. able shows that on the tranng set the coo estmator led to hgher face recognton classfcaton accuracy than the lnear covarance estmator (ooled) and both otmsed quadratc dscrmnant estmators (rda and looc). For the test samles, the rda and looc estmators often outerformed the coo n lower dmensonal sace, but these erformances deterorated when the dmensonalty ncreased, artcularly the looc ones. It seems that n hgher dmensonal sace, when the grou estmate became extremely oorly reresented, the RDA and LOOC arameters dd not counteract the grou mxng sngularty effect. he coo estmator acheved the best recognton rate 96.4% for all PCA comonents consdered. In terms of how senstve the covarance results were to the choce of tranng and test sets, the covarance estmators smlarly had the same erformances, artcularly n hgh dmensonal sace.

12 PCAs ranng grou 99.5(0.4) ooled 73.3(3.) 96.6(.) 99.(0.6) 00.0(0.0) 00.0(0.0) coo 97.0(.) 99.9(0.) 00.0(0.0) 00.0(0.0) 00.0(0.0) rda 8.(.8) 99.5(0.7) 99.9(0.) 00.0(0.0) 00.0(0.0) looc 89.4(.9) 98.9(0.7) 99.6(0.4) 99.8(0.3) 99.9(0.) est grou 5.6(4.4) ooled 59.5(3.0) 88.4(.4) 9.8(.8) 95.4(.5) 95.0(.6) coo 69.8(3.4) 90.(.5) 94.0(.9) 96.4(.6) 95.9(.5) rda 64.7(3.9) 9.4(.9) 94.0(.4) 96.0(.7) 95.6(.6) looc 70.(3.) 90.8(.) 93.5(.) 93.0(.8) 9.0(.8) RDA γ LOOC α α α α α α able. ORL face database results. he results of the ohou facal exresson recognton are resented n table. For more than 0 PCA comonents on the tranng set, the coo estmator erformed better than all the covarance estmators consdered. Regardng the test samles, however, there s no overall domnance of any covarance estmator. In lower dmenson sace, rda led to hgher classfcaton accuraces, followed by coo, ooled and looc. On the other hand, when the dmensonalty ncreased and the true covarance matrces became aarently equal and hghly ellsodal, rda erformed oorly whle coo, ooled and looc mroved. In the hghest dmensonal sace the LOOC otmsaton, whch consders the dagonal form of the ooled estmate, too advantage of the equal-ellsodal behavour (for more than 70 PCAs all α arameters are bgger than the value ) achevng the best recognton rate 87.% for all PCA comonents calculated. In ths recognton alcaton, all the comuted covarance estmators were qute senstve to the choce of the tranng and test sets.

13 PCAs ranng grou 76.3(3.6) ooled 49.6(3.9) 9.9(.7) 97.6(.0) 99.6(0.5) 00.0(0.0) coo 66.6(3.) 95.8(.6) 99.(0.8) 00.0(0.0) 00.0(0.0) rda 75.0(6.7) 96.7(.9) 98.5(.0) 99.(.0) 99.9(0.) looc 5.4(4.9) 9.0(4.) 95.8(.0) 98.8(.3) 99.9(0.3) est grou 38.8(5.6) ooled 6.5(6.8) 70.(7.8) 79.4(5.8) 83.9(7.0) 84.4(6.5) coo 3.5(5.8) 68.3(5.5) 79.5(5.8) 85.0(7.0) 84.(6.0) rda 37.8(5.9) 73.0(7.4) 80.(6.) 79.9(8.7) 8.3(6.7) looc 6.3(5.3) 65.(5.6) 7.(8.) 79.9(8.7) 87.(5.8) RDA γ LOOC α α α α α α able. ohou facal exresson database results. 6 Concluson In ths aer, alternatve otmsed Bayes lug-n covarance estmators avalable n statstcal attern recognton were descrbed, wth regard to the dffcultes caused by small samle szes, and a new covarance estmator was roosed. Exerments were carred out to evaluate these aroaches on two real data recognton tass: face and facal exresson recognton. hese exerments confrmed that choosng an ntermedate estmator between the lnear and quadratc classfers mrove the classfcaton accuracy n settngs for whch samles szes are small and number of arameters or features s large. he new covarance estmator, called Covarance Projecton Orderng method (COPO), has roved to be a owerful technque n small samle sze mage recognton roblems, esecally when concerns about comutatonal costs exst. he new estmator does not requre an otmsaton rocedure, but an egenvector-egenvalue orderng rocessng to select nformaton from the rojected samle grou covarance matrces whenever oss- 3

14 ble and the ooled covarance otherwse. hs estmator can be vewed as a non-sngular aroxmaton of a sngular covarance matrx. he revous statement, however, s not conclusve. Comarsons between estmators le RDA and LOOC have to be analysed consderng other attern recognton roblems. References [And84].W. Anderson, An Introducton to Multvarate tatstcal Analyss, second edton. New Yor: John Wley & ons, 984. [CW98] C.Lu and H. Wechsler, Probablstc reasonng models for face recognton, n Proc. IEEE Comuter ocety Conference on Comuter Vson and Pattern Recognton, anta Barbara, Calforna, UA, June 3-5, 998. [Fr89] J.H. Fredman, Reguralzed Dscrmnant Analyss, Journal of the Amercan tatstcal Assocaton, vol. 84, no. 405, , March 989. [Fu90] K. Fuunaga, Introducton to tatstcal Pattern Recognton, second edton. Boston: Academc Press, 990. [GR89]. Greene and W.. Rayens, Partally ooled covarance matrx estmaton n dscrmnant analyss, Communcatons n tatstcs-heory and Methods, vol. 8, no. 0, , 989. [GR9]. Greene and W.. Rayens, Covarance oolng and stablzaton for classfcaton, Comutatonal tatstcs & Data Analyss, vol.,. 7-4, 99. [HL96] J.P. Hoffbec and D.A. Landgrebe, Covarance Matrx Estmaton and Classfcaton Wth Lmted ranng Data, IEEE ransactons on Pattern Analyss and Machne Intellgence, vol. 8, no. 7, , July 996. [LBA99] M. J. Lyons, J. Budyne, and. Aamatsu, Automatc Classfcaton of ngle Facal Images, IEEE ransactons on Pattern Analyss and Machne Intellgence, vol., no., , December 999. [Ou86] F. O ullvan, A tatstcal Persectve on Ill-Posed Inverse Problems, tatstcal cence, vol., , 986. [GF0] C. E. homaz, D. F. Glles and R. Q. Fetosa, Usng Mxture Covarance Matrces to Imrove Face and Facal Exresson Recogntons, n Proc. of 3 rd Internatonal Conference of Audo- and Vdeo-Based Bometrc Person Authentcaton AVBPA 0, rnger-verlag LNC 09,. 7-77, Halmstad, weden, June 00. [L99]. adjudn and D.A. Landgrebe, Covarance Estmaton Wth Lmted ranng amles, IEEE ransactons on Geoscence and Remote ensng, vol. 37, no. 4, July 999. [P9] M. ur and A. Pentland, Egenfaces for Recognton, Journal of Cogntve Neuroscence, vol. 3,. 7-85, 99. 4