POSSIBILITY FUZZY C-MEANS CLUSTERING FOR EXPRESSION INVARIANT FACE RECOGNITION

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1 POSSIBILITY FUZZY C-MEANS CLUSTERING FOR EXPRESSION INVARIANT FACE RECOGNITION Aruna Bhat Department of Electrcal Engneerng, IIT Delh, HauzKhas, New Delh ABSTRACT Face beng the most natural method of dentfcaton for humans s one of the most sgnfcant bometrc modaltes and varous methods to acheve effcent face recognton have been proposed. However the changes n face owng to dfferent expressons, pose, makeup, llumnaton, age brng about marked varatons n the facal mage. These changes wll nevtably occur and they can be controlled only tll a certan degree beyond whch they are bound to happen and wll affect the face thereby adversely mpactng the performance of any face recognton system. Ths paper proposes a strategy to mprove the classfcaton methodology n face recognton by usng Possblty Fuzzy C-Means Clusterng (PFCM). Ths clusterng technque was used for face recognton due to ts propertes lke outler nsenstvty whch make t a sutable canddate for use n desgnng such robust applcatons.pfcm s a hybrdzaton of Possblstc C-Means (PCM) and Fuzzy C-Means (FCM) clusterng algorthms. PFCM s a robust clusterng technque and s especally sgnfcant for ts nose nsenstvty. It has also resolved the concdent clusters problem whch s faced by other clusterng technques. Therefore the technque can also be used to ncrease the overall robustness of a face recognton system and thereby ncrease ts nvarance and make t a relably usable bometrc modalty. Keywords FCM, PCM, PFCM, EIGEN FACE, FISHER FACE 1. INTRODUCTION Although face recognton comes naturally to all of us, but buldng a computersed system whch s nearly as effcent as human bran s yet a task unaccomplshed though very much achevable. There have been varous face recognton methods proposed n the past but nearly all suffer from the practcal aspect of changes n the face, whch to a certan extent are unavodable. Most of the extng face recognton systems demand precse algnment and correspondence between the testng and the tranng data sets. Ths restrcton makes the system unusable n practce. The changes n the face mage due to varatons n pose, llumnaton, expressons etc. make such strct face recognton systems futle n real world applcatons. The results are even more deplorable f the varatons are collectvely present. However there s no escape to such varatons n real world and there s a need to desgn strong relable face recognton methods. Varous approaches have been proposed for developng an effcent face recognton system. One of the most sgnfcant achevements n the area of face recognton was the Egen Face methodology, proposed by M.A.Turk&A.P.Pentland [1] [2]. It utlzes Prncpal Component Analyss to extract the features from the facal mages [3]. Extendng the Egen face [1] [2] approach further, Hallnan [4] proposed that showed that fve Egen faces were suffcent to represent the face mages under a wde range of varatons. Changng expressons create a lot of DOI: /jc

2 varatons n the face whch makes t very hard to recognze the faces accurately. Extensve work has been done n ths area [5] [6]. Pentland, Moghaddam and Starner [7] proposed a vew based modular Egen space approach for Face Recognton. Zhang et al [8] proposed an approach to fnd features usng Actve Appearance Model (AAM). Vretos et al [9] also used Appearance based approach and SVM classfer wth a reasonably hgh classfcaton accuracy. P. Ekman and E. e. Rosenberg proposed the facal acton codng system (FACS) [10] whch captured the varatons of face features under facal expressons from a sngle mage frame. Unsupervsed technques have also been extensvely studed for ther applcaton n the area of face recognton. Mu-Chun Su and Chen-Hsng Chou suggested a modfed Verson of the K-Means Algorthm for face recognton [11]. Face recognton usng Fuzzy c-means clusterng and sub-nns was developed by Lu J, Yuan X and Yahag T [12]. Clusterng has also found ts use n dynamc and 3D face recognton applcatons too [13] [14]. Extensve research n the feld s n progress [17] so as to acheve a hgh performance face recognton system that can practcally usable n real tme dynamc envronment. Ths paper presents a new approach to perform the classfcaton of faces usng Possblstc Fuzzy C-Means (PFCM) clusterng [15]. Ths clusterng technque was put to test for face recognton due to some of ts propertes lke nose and outler nsenstvty whch make t a sutable canddate for desgnng such robust applcatons. PFCM s a hybrdzaton of Possblstc C-Means (PCM) and Fuzzy C-Means (FCM) clusterng algorthms. It s able to overcome varous problems of PCM, FCM and Fuzzy Possblstc C-Means (FPCM) clusterng algorthms. PFCM s especally sgnfcant as t solved the nose senstvty defect of FCM. It also has resolved the concdent clusters problem of PCM and also elmnated the row sum constrants of FPCM. Therefore t s evdent that the technque s useful for desgnng robust face recognton systems. It can be appled n collaboraton wth theexstng face recognton algorthms such that facal bometrc systems become ncreasngly nvarant to dfferent knds of varatons due to changes n pose, gat, expressons, llumnaton etc. The paper explores the performance of the sad methodology wth respect to the changes n facal expressons whch are known to severely mpact the effcency of face recognton systems. The rest of ths paper s organzed as follows: Secton 2 dscusses the conceptual detals of Possblstc Fuzzy C-Means (PFCM) [15]. Applcaton of PFCM n face recognton s dscussed n detal n the Secton 3. Expermental results have beenelaborated n Secton 4 and Secton 5 dscusses the concluson and the future scope of the proposed technque. 2. OVERVIEW OF METHODOLOGY Clusterng s useful n parttonng of the unlabelled data nto varous clusters. One of the most popular clusterng technques s Fuzzy C-Means (FCM) [18] algorthm where n a data pont s allocated membershp values to varous clusters based on ts relatve dstance tothe data pont prototypesn those clusters representng the cluster centres n the model.however n case a data pont s equdstant from two prototypes, then ts membershp n each one of the clusters wll be the same rrespectve of the absolute value of ts dstance from thetwo centrods as well as from the other data ponts n the data. Ths leads to a major problem n handlng the nose ponts or the outlers. The ssue s that the nose ponts may belocated far away but f they are equdstant fromthe central structure of the two clusters, they may come to have anequal membershp allocated n both. Ideally such nose ponts or outlers should be gven very low / zero membershp n etherof the clusters. 36

3 The objectve functon for FCM s gven by: (1) where, U s the partton matrx, V s a vector of cluster centres, X s a set of all data ponts, x represents a data pont, n s the number of data ponts, c s the number of cluster centers whch are descrbed by s coordnates, x A = x Ax s any nner product norm, u s taken as the membershp of x k n the th parttonng fuzzy subset (cluster) of X. D ka represents the dstance between th cluster and k th data gven by: D ka = x k V A > 0, for all I and k. (2) The membershp value s expressed as: (3) V = (4) A new clusterng algorthm, Possblstc C-Means (PCM) was proposed by Krshnapuram and Keller [19] to counter the aforementoned ssue of nose and outlers. They relaxed the constrant on the membershp, u = 1 for all k,suchthat the sum satsfes a looser constrantwhch s nterpreted as the typcalty of the data pont relatveto a partcular cluster rather than ts membershp n the cluster. Theyrepresented each row of the partton matrx as a possblty dstrbuton over cluster.therefore PCM algorthm ndeed helps n dentfyng outlers and nose ponts. However ths makes PCM s very senstve to ntalzatons whch may sometmes cause t to generate concdent clusters. Also the typcalty can bevery senstve to the values of the varous addtonal parameters neededby the PCM model. The objectve functon for PCM s gven by: (5) 37

4 where, T s the typcalty matrx, V s a vector of cluster centres, X s a set of all data ponts, x represents a data pont, n s the number of data ponts, c s the number of cluster centers whch are descrbed by s coordnates, x A = x Ax s any nner product norm, s taken as the typcalty of x k n the th parttonng fuzzy subset (cluster) of X, γ> 0 s a user defned constant The typcalty matrx T = [ ] cxn has ndependent rows and columns. Therefore mnmzaton of the objectve functon can be done by mnmzng ts j th term wth respect to T. = 1+ /() (6) V = (7) The value of D ka can even be zero here unlke FCM makng PCM overcome the problem of sngularty that FCM suffers from. Therefore wth approprate values of γ and V 0, PCM can sgnfcantly resolve the ssue of outlers and nose ponts. However due to lack of constrants placed on the typcalty matrx, PCM tself may face the ssue of concdent cluster generaton. To deal wth the same Possblty Fuzzy C-Means algorthm was proposed [15]. PFCM whch s a hybrd of FCM and PCM uses the membershp and typcalty aspects from both of the clusterng models and optmzes the followng objectve functon: where, U s the partton matrx, T s the typcalty matrx, V s a vector of cluster centres, X s a set of all data ponts, x represents a data pont, n s the number of data ponts, c s the number of cluster centers whch are descrbed by s coordnates, x A = x Ax s any nner product norm, γ> 0 s a user defned constant s taken as the membershp of x k n the th parttonng fuzzy subset (cluster) of X, s taken as the typcalty of x k n the th parttonng fuzzy subset (cluster) of X. (8) 38

5 It s subject to the constrants u = 1 for all k and u 0 and t 1, a>0, b>0, m>1, ƞ>1, The constants a and b defne the relatve mportance of fuzzy membershp and typcalty values n the objectve functon. By sutable combnaton of these parameters we can make PFCM behave more lke FCM or PCM f and when requred. For PFCM, D ka whch represents the dstance between th cluster and k th data s gven by: Usng the above equaton we can have the followng expressons for the membershp and typcalty values. (9) (10) To reduce the effect of outlers on centrods, we must use a hgher value for b as compared to a and also control the choce of ƞ as per the values of m. (11) Therefore by relaxng the constrant (row sum = 1) on the typcalty values but retanng the column constrant on the membershp values we can acheve the best of both the FCM and PCM algorthmsand acheve sgnfcant robustness aganst the presence of nosy data or outlers. Ths methodology can be appled for the classfcaton of the facal mages n a face recognton system whch s known to be very senstve and prone to outlers, nosy data and even mnute varatons. PFCM can be used for clusterng after feature extracton from the data set. In ths paper the clusterng technque was appled over the data receved from Egen Face [1][2] and Fsher Face [20] methods to evaluate ts performance for face recognton. Clusterng was done over a set of Egen faces and Fsher faces to classfy the face mages of dfferent ndvduals wth varyng expressons. (12) 39

6 3. PROPOSED METHODOLOGY 3.1. Possblty Fuzzy C-Means Clusterng on Egen Faces PFCM was used for clusterng of the data retreved after computnga set of Egen Faces from the data source of facal mages: 1. Obtan face mages I ( = 1,2...M) of same sze, where M = No. of mages n mage data set. 2. Transform all the tranng mages I 1.. to a column vector J Calculate the Mean face m = (J ) (13) 4. Normalse the mages N = J m (14) 5. Obtan the Covarance matrx Cov = N N = AA T (15) 6. Derve the egenvectors for the matrx A T A of sze m x m. 7. If v s a nonzero vector and λ s a number such that Av = λv, then v s sad to be an egenvector of A wth egenvalue λ. 8. Consder the egenvectors v of A T A: A T Av = µ v (16) 9. Pre-multplyng both sdes by A such that, AA T (Av ) = µ (Av ) (17) 10. The egenvectors of covarance matrx are u = Av (18) 11. Face space s gven by F = u T N (19) 12. Perform the Possblty Fuzzy C-Means clusterng over the set N, for P = 1 to c, where c s no. of classes (or subjects) and mnmse the followng objectve functon as mentoned n equaton 8, where, U s the face class partton matrx, T s the face typcalty matrx, V s the vector of cluster centers, X s thenput set ofegen Faces fromf = u T N, x represents a data pont, n s the number of data ponts, c s the number of cluster centers, 40

7 X, x A = x Ax s any nner product norm, γ> 0 s a user defned constant s taken as the membershp of x k n the th parttonng fuzzy subset (cluster) of s taken as the typcalty of x k n the th parttonng fuzzy subset (cluster) of X, u = 1 for all k and u 0 and t 1, a>0, b>0, m>1, ƞ>1, 13. Fnd the dstance between th cluster and k th data tem, membershp matrx and typcalty matrx and updated values of cluster means as explaned n equaton 9,10 and Generate P clusters C 1.. C P. Obtan the optmal value of the membershp for each face mage among all clusters, for P = 1 to c. 15. All the face mages are assgned to a partcular cluster based on whether they match the membershp threshold condton followng whch theywll be correctly assgned to the partcular cluster representng the face of an ndvdual subject Possblty Fuzzy C-Means Clusterng on Fsher Faces PFCM was also evaluated for clusterng the data retreved after computng a set of Fsher Faces from the data source of facal mages: 1. X 1, X 2,,X c represent the c classes of face n the database. Each face class s represented by vector X, = 1, 2,, c has k facal mages x j, j=1, 2,, k. 2. Fnd the mean mage µ of each class X as: 3. Mean mage µ of all the face classes n the database can be calculated as: 4. Calculate the wthn-class scatter matrx: 1 µ = 5. Calculate the between-class scatter matrx: S c B = N = 1 S ( µ µ )( µ µ ) 6. We fnd the projecton drectons as the matrx W that maxmzes (20) (21) (22) (23) Ẑ = argmaxj(z) = Z T S B Z / Z T S W Z (24) 7. We can observe that t represents a generalzed egenvalue problem where the columns of Z are gven by the vectors Z such that, 8. S B Z = λ S W Z (25) 9. Fnd the product of S W -1 and S B and calculate ts egenvectors after reducng the dmenson of the feature space. 10. Compose a matrx Z that represents all egenvectors of S W -1 * S B by placng each egenvector Z as a column n Z. k x j k j= 1 1 µ = c W = ( = 1 xk X c c = 1 x k µ µ )( x µ ) k T T 41

8 11. Each face mage x j X can be projected nto ths face space as: F = Z T (x j µ) (26) 12. Perform the Possblty Fuzzy C-Means clusterng over the set F, for P = 1 to c, where c s no. of classes (or subjects) and mnmse the followng objectve functon as mentoned n equaton 8, where, X, U s the face class partton matrx, T s the face typcalty matrx, V s the vector of cluster centers, X s thenput set offsher Faces from F = u T N, x represents a data pont, n s the number of data ponts, c s the number of cluster centers, x A = x Ax s any nner product norm, γ> 0 s a user defned constant s taken as the membershp of x k n the th parttonng fuzzy subset (cluster) of s taken as the typcalty of x k n the th parttonng fuzzy subset (cluster) of X, u = 1 for all k and u 0 and t 1, a>0, b>0, m>1, ƞ>1, 13. Fnd the dstance between th cluster and k th data tem, membershp matrx and typcalty matrx and updated values of cluster means as explaned n equaton 9,10 and Generate P clusters C 1.. C P. Obtan the optmal value of the membershp for each face mage among all clusters, for P = 1 to c. 15. All the face mages are assgned to a partcular cluster based on whether they match the membershp threshold condton followng whch they wll be correctly assgned to the partcular cluster representng the face of an ndvdual subject. The resultant clusterng classfes the face mages of subjects wth varyng degrees of expressons to ther correct ndvdual classes. 4. EXPERIMENTAL RESULTS Open source data sets were used to evaluate the performance of the technque. JAFFE database [25] contans the face mages wth varyng expressons and emotons. The face mages were segmented and pre-processed followng whch Egen Face and Fsher Face algorthms were appled.clusterng of the data set thus retreved was done usng Possblty Fuzzy C-Means clusterng technque whch was found to be very effectve n classfyng the facal mages correctly even n the presence of varatons n the mages owng the presence of outlers, nose and changes n expressons and emotons. The results were compared to the methods of Egen Face and Fsher face. When combned wth PFCM both these methods performed sgnfcantly better and yelded favourable results wth more robustness towards varatons n expressons. Thus use of PFCM was found to amplfy the nvarance of the face recognton system and make 42

9 t sturder. Therefore t encourages the applcaton of PFCM n combnaton wth varous standard and upcomng face recognton methodologes to boost ther overall performance even n the presence of varatons n the frontal mage of the face lke the changes n expressons. The technque can be extended further to allow ts collaboraton wth dfferent methodologes that ad n formng strong face recognton algorthms. The results obtaned clearly depct an ncrease n the recognton accuracy n spte of the presence of nose, outlers and varatons n expressons n the face mages when the PFCM s used wth standard methods of Egen Face and Fsher Face as shown below: Egen Face Egen Face+PFCM Fgure 1.Performance of Egen Face vs. PFCM+Egen Face Fsher Face Fsher Face+PFCM Fgure 2.Performance of Fsher Face vs. PFCM+Fsher Face 43

10 5. CONCLUSIONS AND FUTURE SCOPE The propertes of outler and nose nsenstvty of Possblty Fuzzy C-Means clusterng have contrbuted to ts use n desgnng a strong face recognton system n ths paper. Increase n performance of face recognton by applyng PFCM wth standard technques of Egen Face and Fsher Face leads us towards ts applcaton n desgnng robust face recognton systems. As observed through the expermental analyss, PFCM has made sgnfcant amplfcaton n the success rate of the face recognton even n presence of the outlers, nose and changes n expressons. The clusterng technque may be analysed and studedfurther for f t can also contrbute to desgnng face recognton applcatons that are nvarant to changes n pose, llumnaton, facal dstractons lke makeup and har growth. Facal varatons are not lmted to changes n expressons. In fact dfferent lght condtons and posture changes make extreme varatons n the face mage whch are more dffcult to deal wth as well. These changes are bound to occur n practcal scenaros therefore PFCM needs to be tested and mproved f requred for caterng to nullfyng the effect of such changes to the best possble extent. The technque may be used to work n collaboraton wth varous other methodologes for reducng the mpact of such changes. ACKNOWLEDGEMENTS The author would lke to thank the faculty and the staff at Indan Insttute of Technology, Delh for thervalued gudance and help they have gracously provded for the research work. REFERENCES [1] M.Turk&A.Pentland, (1991) Egenfaces for Recognton, Journal of Cogntve Neuroscence, vol.3, no.1, pp , 1991a. [2] M. Turk & A. Pentland, (1991) Face Recognton Usng Egenfaces, Proc. IEEE Conf. on Computer Vson and Pattern Recognton, pp [3] P.J.B. Hancock, A.M. Burton & V. Bruce, (1996) Face processng: Human percepton and prncpal components analyss, Memory and Cognton, 24(1):26-40 [4] P. Hallnan. A low-dmensonal representaton of human faces for arbtrary lghtng condtons. In Proceedngs. IEEE Conference. CVPR, [5] Rothkrantz M., Automatc Analyss of Facal Expressons: The State of the Art, IEEE Transacton Pattern Analyss and Machne Intellgence, vol. 22, no. 12, pp , [6] Fasel B. and Luettn J., Automatc Facal Expresson Analyss: a Survey, IEEE Pattern Recognton, vol. 36, no. 1, pp , [7] Pentland A., Moghaddam B., and Starner T., Vew-Based and Modular Egenspaces for Face Recognton, n Proceedngs of IEEE Conference on Computer Vson and Pattern Recognton, pp.84-91, [8] Zhang Y. and Martnez A., Recognton of Expresson Varant Faces Usng Weghted Subspaces, n Proceedngs of the 17th Internatonal Conference on Pattern Recognton, vol. 3, pp , [9] Vretos N., Nkolads N., and Ptas I., A Model-Based Facal Expresson Recognton Algorthm usng Prncpal Component Analyss, n Proceedngs of the 16th IEEE Internatonal Conference on mage Processng, Caro, pp , [10] P. Ekman and E. e. Rosenberg. The facal acton codng system: A technque for the measurement of facal movement. Consultng Psychologsts press, [11] Mu-Chun Su and Chen-Hsng Chou, (June 2001)A Modfed Verson of the K-Means Algorthm wth a Dstance Based on Cluster Symmetry, IEEE Transactons On Pattern Analyss And Machne Intellgence, Vol. 23, No. 6. [12] Lu J, Yuan X, Yahag T., (2007 Jan) A method of face recognton based on fuzzy c-means clusterng and assocated sub-nns, IEEE Transactons Neural Networks, 18(1):

11 [13] MarosKyperountasa, Author Vtae, AnastasosTefasa, Author Vtae, Ioanns Ptas, (March 2008) Dynamc tranng usng multstage clusterng for face recognton Pattern recognton, Elsever, Volume 41, Issue 3, Pages [14] Zheng Zhang, Zheng Zhao, Gang Ba, (2009) 3D Representatve Face and Clusterng Based Illumnaton Estmaton for Face Recognton and Expresson Recognton, Advances n Neural Networks ISNN 2009 Lecture Notes n Computer Scence Volume 5553, pp [15] Nkhl R. Pal, Kuhu Pal, James M. Keller, and James C. Bezdek. (2005), "A Possblstc Fuzzy c- Means Clusterng Algorthm", IEEE Transactons on Fuzzy Systems, Vol. 13, No. 4, August [16] JAFFE database, Codng Facal Expressons wth Gabor Wavelets Mchael J. Lyons, Shgeru Akamatsu, Myuk Kamach&JroGyoba Proceedngs, Thrd IEEE Internatonal Conference on Automatc Face and Gesture Recognton, Aprl , Nara Japan, IEEE Computer Socety, pp [17] W. Zhao, R. Chellappa, A. Rosenfeld, P.J. Phllps, Face Recognton: A Lterature Survey. [18] Bezdek, James C. (1981). Pattern Recognton wth Fuzzy Objectve Functon Algorthms. [19] R. Krshnapuram and J. Keller, A possblstc approach to clusterng, IEEE Trans. Fuzzy Syst., vol. 1, no. 2, pp , Apr [20] P.N. Belhumeur, J.P. Hespanha& D.J. Kregman, (July 1997) Egenfaces vs. Fsherfaces: Recognton Usng Class-Specfc Lnear Projecton, PAMI(19), No. 7, pp Authors The author s a research scholar at Indan Insttute of Technology, HauzKhas, Delh. 45