Comparative Study between different Eigenspace-based Approaches for Face Recognition

Transcription

1 Coparatve Study between dfferent Egenspace-based Approaches for Face Recognton Pablo Navarrete and Javer Ruz-del-Solar Departent of Electrcal Engneerng, Unversdad de Chle, CHILE Eal: {pnavarre, Abstract. Dfferent egenspace-based approaches have been proposed for the recognton of faces. They dffer ostly n the knd of projecton ethod been used and n the slarty atchng crteron eployed. The a of ths paper s to present a coparatve study between soe of these dfferent approaches. Ths study consders theoretcal aspects as well as sulatons perfored usng a face database wth a few nuber of classes. 1 Introducton Aong the ost successful approaches used n face recognton we can enton egenspace-based ethods, whch are ostly derved fro the Egenface-algorth. These ethods project the nput faces onto a densonal reduced space where the recognton s carred out, perforng a holstc analyss of the faces. Dfferent egenspace-based approaches have been proposed. They dffer ostly n the knd of projecton/decoposton ethod been used and n the slarty atchng crteron eployed. The a of ths paper s to present a coparatve study between soe of these dfferent approaches. The coparson consders the use of three dfferent projecton ethods (Prncpal Coponent Analyss, Fsher Lnear Dscrnant and Evolutonary Pursut) and fve dfferent slarty atchng crtera (Eucldean-, Cosnes- and Mahalanobs-dstance, Self-Organzng Map and Fuzzy Feature Contrast). The pre-processng aspects of these approaches (noralzaton, llunaton nvarance, geoetrcal nvarance, etc.) are not gong to be addressed n ths study. It should be noted that a prevous coparatve study that does not nclude the Fuzzy Feature Contrast ethod was presented n [4]. The entoned ethods are descrbed n secton 2, and the coparatve study s presented n secton 3. 2 Egenspace-based Approaches Egenspace-based approaches approxate the face vectors (face ages) wth lower densonal feature vectors. The an supposton behnd ths procedure s that the face space (gven by the feature vectors) has a lower denson than the age space (gven by the nuber of pxels n the age), and that the recognton of the faces can be perfored n ths reduced space. These approaches consder an off-lne phase or tranng, where the face database s created and the projecton atrx, the one that acheve the densonal reducton, s obtaned fro all the database face ages. In the off-lne phase are also calculated the ean face and the reduced

2 representaton of each database age. These representatons are the ones to be used n the recognton process. 2.1 General Approach Fgure 1 shows the block dagra of a generc egenspace-based face recognton syste. A preprocessng odule transfors the face age nto a untary vector and then perfors a subtracton of the ean face ( x). After that, the resultng vector, x, s projected usng the projecton atrx W R N that depends on the egenspace ethod been used (see secton 2.2). Ths projecton corresponds to a densonal reducton of the nput, startng wth vectors x n R N (wth N the age vector denson) and obtanng projected vectors q n R wth <N (usually <<N). The Slarty Matchng odule copares the slarty of the reduced representaton of the query face vector q wth the reduced vectors p k R that represent the faces n the database. By usng a gven crteron of slarty (see secton 2.3), ths odule deternes the ost slar vector p k n the database. The class of ths vector s the result of the recognton process,.e. the dentty of the face. In addton, a Rejecton Syste for unknown faces s used f the slarty atchng easure s not good enough (see descrpton n [1]). Fg. 1. Block dagra of a gven egen-space face recognton syste. 2.2 Projecton/Decoposton Methods Prncpal Coponents Analyss - s a general ethod to dentfy the prncpal dfferences between sgnals and after that to ake a densonal reducton of the. Let X= ( x 1 x)( x 2 x) L( x NT x) be the atrx of the noralzed tranng vectors. [ ] x j represents the noralzed j age vector, x s the ean face age and NT s the nuber of tranng ages. Then, R = XX T wll be the correlaton atrx estator. The egenvectors of R represent a specal bass n the age space, and the egenvalues are the projecton varance on each of ths axes (the Egenfaces). Therefore wll chose only the egenvectors of R assocated wth the hgher

3 varance and n ths way wll reduce the denson of the tranng ages. Also gve us the projecton atrx W R N for reducng every age that follows the sae statstcal pattern. Coputatonal aspects of the pleentaton of ths ethod are explaned n [4]. Fsher Lnear Dscrnant - FLD FLD searches for the projecton axes on whch the face ages of dfferent classes are far fro each other, and at the sae te where the face ages of the sae class are close fro each other. In a slar way of usng the R atrx, FLD uses two scatter atrces, S b and S w, for representng the separaton between the ndvdual class eans respect to the global ean face, and the separaton between vectors of each class respect to ther own class ean, respectvely: NC ( )( ) NC ( )( ) Sb = P T () () ( C) ; S x x w = P C E T () () () () ( ) (1) = 1 = 1 where s the global ean vector, P( C ) are the occurrence probabltes assocated to each class C, () are the average vectors of C, and x () are the vectors assocated to C. The axzaton of the between class scatter and the nzaton of the wthn class scatter s perfored by solvng the general egensyste k k Sbw = λ k Sw w. The resultng non-orthonoral base represents the projecton atrx W R N, where the rows are the general egenvectors assocated wth the largest general egenvalues (Fsher Paraeters [4]). To solve the proble of the large sze of the scatter atrces, s appled before FLD. In ths way we are also solvng the proble of sngularty for S w. Evolutonary Pursut - EP EP, orgnally proposed n [3], searches for the best set of projecton axes n order to axze a ftness functon that easures, at the sae te, the classfcaton accuracy and generalzaton ablty of the syste. Because the denson of the soluton-space of ths proble s very large, t s solved usng Genetc Algorths. In order to obtan the EP-faces an ntal densonal reducton s frst perfored usng, and then a Whtenng Transforaton s appled (equvalent to a Mahalanobs etrc syste, see 2.3). In the Whtened- space are perfored several rotatons between par of axes and then a subset of the s selected. Each rotaton s coded usng a chroosoe representaton. In ths representaton each chroosoe represents a certan projecton syste. To evaluate ths syste the followng ftness functon s used: ζ( α, a ) = ζ ( α, a ) + λζ ( α, a ) (2), k a k s k where ζa( αk, a) easures the accuracy, ζs( αk, a) easures the generalzaton ablty, and λ s a postve constant (see defntons n [3]).

4 2.3 Slarty Matchng Methods Eucldean Dstance d( x, y) = ( x y) ( x y) T. (3) Cosne Dstance cos( xy, ) = T x y x y. (4) Mahalanobs Dstance ( ) ( ) T 1 d( xy, ) = x y R x y ; R: correlaton atrx. (5) Fro a geoetrcal pont of vew ths dstance has a scalng effect n the age space. Takng nto consderaton the face age subset, drectons n whch a greater varance exst are copressed and drectons n whch a saller varance exst are expanded. It can be proved that n the space the Mahalanobs dstance s equvalent to the Eucldean dstance, weghtng each coponent by the nverse correspondent egenvalue (see deonstraton n [4]), and t s often called Whtenng () Transforaton. SOM Clusterng Self-Organzng Maps (SOMs) are used as assocatve networks to atch the projected query face wth the correspondng projected database faces. The use of a SOM to pleent ths odule proves the generalzaton ablty of the syste. The SOM approach uses reference vectors to approxate the probablty dstrbuton of the faces n a 2D ap [2]. In the tranng phase of the SOM a clusterng of the reduced face vectors s carred out. Thereafter the SOM s transfored n an assocatve network by labelng all ts nodes. Both procedures are explaned n [4]. Fuzzy Feature Contrast Ð FFC S ( x, y) = n { µ ( x), µ ( y) } α ax { µ ( x) µ ( y), 0} β ax { µ ( y) µ ( x), 0 } (6) = 1 = 1 = 1 where µ (x) s a ebershp functon assocated wth the -coponent of vector x R. Ths slarty easure, orgnaly proposed n [5], s a fuzzy pleentaton of the Feature Contrast odel fro Tversky. The frst su easure the coon features (ntersecton) and the others represent the dstnctve features (dfference n the two possble ways). The postve paraeters α and β adjust the contrast of the three knd of features. By chosng α β t s possble to ntroduce asyetres between

5 the subject-referent coparson. Ths odel consders that all the features are ndependent, and that can be assued n and W, but not n FLD and EP. In our pleentaton we noralze each feature of (n W t s not necessary) and we chosed µ (x) lnear between Ð1 and 1, wth x noralzed. 3 Coparson aong the approaches In order to test the descrbed ethods we have ade several sulatons based n the Yale Unversty - Face Iage Database. We use 150 ages of 15 dfferent classes. Then we preprocessed the ages by askng the n wndows of 100 x 200 pxels placng the several face features n the sae relatve places. In table 1 we show the results of several sulatons usng dfferent knd of representatons and slarty atchng ethods. For each sulaton we used a fxed nuber of tranng ages, usng the sae type of ages per class, accordng wth the Yale database specfcaton. In order to obtan representatve results we take the average of 20 dfferent set of ages for each fxed nuber of tranng ages. All the ages not used for tranng are used for testng. We can see that the best odels always are obtaned wth the Fsher representaton, and the dfference aganst the other representatons decrease when the nuber of tranng ages per class decrease, showng that the FLD dscrnaton ablty strongly depends on the nuber of tranng ages per class. The best results are alost always obtaned wth FLD- cosne. The systes that see to be as effcent as FLD-cosne are SOM and Wthenng-cosne. The best results usng FFC were obtaned eployng an asyetrc subject-referent coparson: α=0.5 β=5. Ths eans that n the queston Òhow s the subject face slar to the referent face?ó the answer focus ore on the features of the referent (the unknown face). The generalzaton ablty of the systes s not well easured n our sulatons because the nuber of selected axes s about the sae of the nuber of classes (15). That affects the FLD representaton ethod as well as the FFC and SOM slarty atchng ethods. For ths reason n future works we want to perfor our coparatve study on a larger database, lke FERET. We thnk that ths wll prove the relatve recognton ablty of the ethods beng affected for the sall nuber of classes. Another portant ssue s the coputatonal cost of the tranng processes. In ths coputatonal cost s anly due to the process of deternng R, O( NT 2 N), and solvng the egensyste, O ( NT 3 ). If we suppose that the nuber of tranng ages NT s uch saller than the nuber of pxels per age N, then the coputatonal cost of s just the cost of deternng R, O( NT 2 N). In our pleentaton of FLD we requres prevously the coputaton of to reduce the vectors denson to 1 (1 < NT), and the addtonal cost s due to the process of deternng the scatter atrces, O ( 1 2 NC), and solvng the general egensyste, O ( 1 3 ). Nevertheless the addtonal cost n FLD s usually uch saller than the ntal cost. Fnally EP requres uch ore coputatons because ths process ust terate untl a gven crteron s accoplshed. The coputatonal cost of on-lne operaton s anly gven by the coparsons wth database vectors, O( NT ),

6 except when the SOM-based slarty easure s used, O(( nuber of nodes) ). The nuercal stablty for the dfferent ethods depends ostly of the nuercal algorths used for solvng egensystes. Ether n or FLD ths s not a crtcal proble because always nvolves the anageent of syetrc atrces. Table 1. Mean recognton rates usng dfferent nubers of tranng ages per class, and takng the average of 20 dfferent tranng sets. The sall nubers are the standard devaton of each recognton rate. Whtenng Whtenng Whtenng Whtenng./class axes Eucldean cos(?) SOM FFC Eucldean cos(?) SOM FFC FISHER FISHER FISHER FISHER FISHER Acknowledgeents Ths research was supported by the DID (U. de Chle) under Project ENL-2001/11 and by the jon "Progra of Scentfc Cooperaton" of CONICYT (Chle) and BMBF (Gerany). References 1. Golfarell M., Mao D., and Malton D., ÒOn the Error-Reject Trade-Off n Boetrc Verfcaton SystesÓ, IEEE Trans. Pattern Analyss and Machne Intellgence, vol. 19, no. 7, pp , July Kohonen T., ÒSelf-Organzed MapsÓ, 1997.

7 3. Lu C., and Wechsler H., ÒEvolutonary Pursut and Its Applcaton to Face RecogntonÓ, IEEE Trans. Pattern Analyss and Machne Intellgence, vol. 22, no. 6, pp , June Navarrete P., and Ruz del Solar J., ÒEgenspace-based Recognton of Faces: Coparsons and a new ApproachÓ, Proc. of the Int. Conf. on Iage Analyss and Processng ICIAP 2001, pp , Sept , Palero, Italy. 5. Santn S., and Jan R., ÒSlarty MeasuresÓ, IEEE Trans. Pattern Analyss and Machne Intellgence, vol. 21, no. 9, pp , Septeber 1999.