LDA-based Non-negative Matrix Factorization for Supervised Face Recognition

Transcription

1 1294 JOURNAL OF SOFTWARE, VOL. 9, NO. 5, MAY 2014 LDA-based No-egative Matrix Factorizatio for Supervised Face Recogitio Yu Xue a, Chog Sze Tog b, Jig Yu Yua c a School of Physics ad Telecommuicatio Egieerig, South Chia Normal Uiversity, Guagzhou Guagdog , Chia xueyu@scu.edu.c b Departmet of Mathematics, Hog Kog Baptist Uiversity, Hog Kog, Chia cstog@math.hkbu.edu.hk c Departameto de Matematica - UFPR,Cetro Politecico, CP:19.081, , Curitiba, PR, Brazil ji@mat.ufpr.br Abstract I PCA based face recogitio, the basis images may cotai egative pixels ad thus do ot facilitate physical iterpretatio. Recetly, the techique of oegative matrix Factorizatio (NMF) has bee applied to face recogitio: the o-egativity costrait of NMF yields a localized parts-based represetatio which achieves a recogitio rate that is o par with the eigeface approach. I this paper, we propose a ew variatio of the NMF algorithm that icorporates traiig iformatio i a supervised learig settig. We itegrate a additioal term based o Fisher s Liear Discrimiat Aalysis ito the NMF algorithm ad prove that our ew update rule ca maitai the o-egativity costrait uder a mild coditio ad hece preserve the ituitive meaig for the base vectors ad weight vectors while facilitatig the supervised learig of withi-class ad betwee-class iformatio. We tested our ew algorithm o the well-kow ORL database, CMU PIE database ad FERET database, ad the results from experimets are very ecouragig compared with traditioal techiques icludig the origial NMF, the Eigeface method, the sequetial NMF+LDA method ad the Fisherface method. Idex Terms oegative matrix factorizatio, pricipal compoet aalysis, fisher liear discrimiat aalysis, eigeface, fisherface I. INTRODUCTION As a importat problems of image processig, face recogitio has received sigificat attetio. It is a typical patter recogitio problem which ca be applied to promote the resolutio of may other classificatio problems, ad also has lots of potetial applicatios. Geerally, automatic face recogitio cotais two approaches, amely, costituet-based ad face-based methods [1], [2]. The first approach performs recogitio based o the spatial relatioship betwee differet facial features such as eyes, mouth ad ose [3], [4], therefore its performace is highly sesitive to the accuracy of facial feature detectio algorithms. However, sice extractig facial compoets accurately is difficult, a small error at this stage may cause a large classificatio error. I cotrast, the face-based approach defies the face as a whole [5], [6], i.e. the correspodig image is cosidered as a two-dimesioal patter ad classified based o its uderlyig statistics. As a effective face-based approach, Pricipal Compoet Aalysis (PCA) [5] uses the orthogoal basis images which have a statistical iterpretatio as the directios of largest variace. Based o this idea, Turk ad Petlad [6] built a face recogitio system i which the sigificat features (eigevectors associated with large eigevalues) are called eigefaces. This approach represets a face image usig a weighted sum of eigefaces ad the classificatio is performed by comparig the weight vectors of the test images with those of the referece face images. However, the traditioal Eigeface-based methods suffer from some limitatios. Firstly, it is well kow that PCA gives a very good represetatio of the images. Give two images of the same perso, the similarity measured uder PCA represetatio is very high. However, for the two images of differet persos, the similarity measured is still high. This suggests this method may ot ejoy a very high discrimiatory ability. The secod problem is that may of the basis images do ot have a obvious visual iterpretatio. This is because PCA allows the coefficiets of base vectors ad weight vectors to be of arbitrary sig. Sice the basis images are used i liear combiatios that geerally ivolve complex cacellatios betwee positive ad egative umbers, may of them do ot have ituitive meaig. Fially, the PCA approach is based o extractig global face features, so the case of occlusios is difficult to hadle. Recetly, a ew techique to obtai a liear represetatio of data has bee proposed. This ew method, called o-egative matrix factorizatio (NMF), was first used i the work of Lee ad Seug [7] to fid parts of objects. NMF differs from other methods by the usage of oegativity costraits. Recetly NMF has bee widely applied i may fields, ad it has bee show to outperform PCA i image recogitio for certai face databases [8] [10]. NMF factorizes the image database ito two matrix factors, whose etries are all o-egative, ad produces a parts- doi: /jsw

2 JOURNAL OF SOFTWARE, VOL. 9, NO. 5, MAY based represetatio of images because it allows oly additive, ot subtractive, combiatios of basis images. Therefore the resultat NMF bases are localized features which correspod to the ituitive idea of combiig parts to form a whole. I real face databases, the images geerally cotai atural occlusios such as suglasses ad scarfs. If some such images are cotaied i the traiig set, the the basis images would be sigificatly affected i global methods such as PCA while the local basis images of NMF remai relatively stable ad thus obtai better recogitio performace (see eg [8]). However, sice the traditioal NMF method is a usupervised learig techique, it s difficult to take advatage of the discrimiatio iformatio i the traiig set to boost the classificatio capability. I this paper we itroduce a LDA-based No-egative Matrix Factorizatio algorithm which is a ew variatio to NMF. To take advatage of more iformatio i the traiig images, we add the Fisher Liear Discrimiat ito the objective fuctio i NMF algorithm, which will lead to more discrimiatory base vectors ad weight vectors. Sice this algorithm ecodes discrimiatio iformatio for face recogitio, it ca improve the result for classificatio. As our ew approach is based o a subspace defiitio, we have used the Pricipal Compoet Aalysis (PCA), the origial NMF algorithm ad the Fisherface method for direct compariso. Moreover, we also use NMF as a method of reducig dimesios ad the subject the projected vectors to LDA to extract the feature vectors for face recogitio. Sice this sequetial NMF+LDA process would preserve the usupervised ature of N- MF as iitially formulated by Lee ad Seug, a direct compariso betwee this procedure ad our algorithm is importat. After implemetig all these methods o the face image databases, the results from experimets support the coclusio that our ew algorithm ca achieve a better performace i face recogitio. This paper is orgaized as follows. Sectio 2 reviews the backgroud of PCA, NMF ad Fisherface. The details of our LDA-based No-egative Matrix Factorizatio algorithm are described i sectio 3. All the testig databases used i this paper are described i sectio 4. Results from experimets o a face recogitio system based o the proposed method are discussed i sectio 5. Coclusios & future work are preseted i sectio 6. A earlier versio of this paper was preseted at the 18th Iteratioal Coferece o Patter Recogitio, 2006 [11]. I this paper, a detailed aalysis ad proof about our model is provided, ad we use more relevat algorithms for the performace compariso based o some well kow face databases. II. REVIEW OF PCA, NMF AND FISHERFACE FOR FACE RECOGNITION This sectio provides the backgroud theory of PCA, NMF, ad Fisherface for face recogitio. A. Pricipal compoet aalysis PCA is origially used to fid a low dimesioal represetatio of data [12]. Some details are described as follows. Let X = {X R d = 1,..., N} be a set of vectors. I image processig, they are formed by row or colum cocateatio of the origial image matrix, with d beig the product of the width ad the height of a image. Let E[X] = 1 N X N =1 be the mea vector i the image set. After subtractig it from each vector of X, we get ˆX = { ˆX, = 1,..., N} with ˆX = X E[X] The we defie the auto-covariace matrix M for X as M = cov(x) = E( ˆX ˆX) where M R d d, with elemets M(i, j) = 1 N 1 N ( ˆX (i) ˆX (j)), 1 i, j d =1 From matrix theory, it s well kow that the autocovariace matrix for the eigevectors is diagoal, it follows that the coordiates of the vectors i X with respect to the eigevectors are u-correlated radom variables. The the PCA of a vector y ca be calculated by projectig it oto the subspace which is spaed by d eigevectors correspodig to the top d eigevalues of the autocorrelatio matrix M sorted i descedig order. Furthermore, Eigefaces are the eigevectors associated with the largest eigevalues from the PCA method. After represetig a face image usig a weighted sum of eigefaces, face recogitio is performed by comparig the correspodig weight vectors betwee the test image ad referece faces. B. NMF method No-egative Matrix Factorizatio (NMF) [7] is a method to obtai a represetatio of data uder oegativity costraits. These costraits produce a partsbased represetatio because they allow oly additive, ot subtractive, combiatios, so the correspodig basis images ca be uderstood as localized features that correspod better with ituitive otios of the parts of face images. If a iitial image database is represeted as a m matrix V, where each colum is a o-egative vector correspodig to a face image, we ca fid two ew oegative matrices (W ad H) to approximate the origial data matrix r V ij (W H) ij = W ia H aj, W R r, H R r m a=1 (1)

3 1296 JOURNAL OF SOFTWARE, VOL. 9, NO. 5, MAY 2014 Each colum of matrix W represets a basis vector while each colum of H represets the weights used to approximate the correspodig colum i V usig the bases from W. For the NMF method, i cotrast to PCA, o subtractios ca occur, so the o-egativity costraits are compatible with the ituitive idea of combiig parts to form a whole face, which is how NMF lears a partsbased represetatio. The depedecies betwee image pixels ad the ecodig variables are depicted i Fig.1. The top odes represet a ecodig h 1,..., h r (colum of H), ad the bottom odes represet a image v 1,..., v (colum of V ). The oegative value W ia characterizes the extet of ifluece that the a th ecodig variable h a has o the i th image pixel v i. Because of the o-egativity of W ia, the image pixels i the same part of face image will be coactivated whe the part is preset, ad NMF lears by adaptig W ia to geerate the optimal approximatio. N o R ep r es e t a im a ge d a ta b a s e a s a m a tr ix V I itia lize th e m a tr ices W a d H F ix W a d u p d a te H F ix H a d u p d a te W C h eck for co ver ge ce Y es Figure 2. Flowchart of the NMF algorithm (after [14]). Figure 1. Probabilistic hidde variables model uderlyig o-egative matrix factorizatio [7]. The update rule for NMF is derived as below: First defie a objective fuctio to measure the similarity betwee V ad W H: F = i=1 j=1 m V ij [V ij log V ij + (W H) ij ] (2) (W H) ij The a iterative algorithm to reach a local maximum of this objective fuctio is give [7]: W t+1 ia = Wia t V ij (W t H t Haj t (3) ) j ij ia = H t+1 aj ia j = H t aj ja i ia V ij (4) ( H t ) ij (5) The covergece of the process is proved i [13]. The flow chart of the NMF algorithm is as below. I face recogitio, NMF is applied as follows. Firstly, i the feature extractio stage, all the traiig images form the origial data matrix V, the the bases W 1 ca be obtaied usig the above update rules (3)-(5). Next, let W + = (W1 T W 1 ) 1 W1 T, the each traiig face image V t is projected oto the liear space as a feature vector H t = W + V t which is the used as a prototype feature poit. A probe image V p to be classified is represeted as H p = W + V p. Fially, we classify the probe images usig the earest eighbour classificatio scheme. At this stage, some suitable distace betwee the weight vectors of the probe image ad traiig image, dist(h p, H t ), is calculated, the the probe image is classified to the class with the miimum distace. C. Fisherface ad sequetial NMF+LDA As a classical techique i multivariate statistics, LDA has also bee successfully applied i patter recogitio [15]. It utilizes class-specific iformatio withi the traiig images ad fids a feature space i which the ratio of the betwee-class scatter ad the withi-class scatter is maximized. However, for the face recogitio problem, the withiclass scatter matrix is ofte sigular. I order to overcome this difficulty, Peter N. Belhumeur, Joao P. Hespaha, ad David J. Kriegma [16] proposed the Fisherface method, which combies PCA with the stadard LDA method to reduce the dimesioality of feature vectors. Nowadays, it is oe of the most popular feature extractio ad dimesio reductio techiques used i the face-based approach.

4 JOURNAL OF SOFTWARE, VOL. 9, NO. 5, MAY I the Fisherface procedure, oe of the class iformatio is icorporated i the PCA stage, while all the PCAbased feature vectors are subsequetly used to extract the LDA-based feature vectors so that the classificatio is more robust. Similarly, we could also first use NMF without ay class iformatio to reduce the dimesioality of feature vectors, ad the use the LDA method to take advatage of the class iformatio i the same way. Sice this sequetial NMF+LDA process is a simple procedure to preserve the ature of NMF as iitially formulated by Lee ad Seug [7], we will use it ad Fisherface as bechmarks for compariso with our LDA-based NMF method, which also exploits the same class iformatio withi the traiig images. A. Our model III. LDA-BASED NMF I the origial NMF model ad Eigeface model, the traiig face images are used collectively without referece to the class membership of the traiig faces. To improve the performace for face recogitio, we propose to add the Fisher discrimiat to the origial NMF algorithm formulatio. Because the colums of the weight matrix H have a oe-to-oe correspodece with the colums of the image matrix V, we aturally hope to maximize the betwee-class scatter ad simultaeously miimize the withi-class scatter of H. Based o this idea, we defie the ew objective fuctio: m V ij F 1 = [V ij log V ij + (W H) ij ] (W H) ij (6) i=1 j=1 + αs W αs B, where α > 0 is a regularizatio parameter, S W is the withi-class scatter of the weight matrix H, ad S B is the betwee-class scatter of H. Let M tr deote the umber of vectors i each class ad C deote the umber of classes. We defie S W ad S B as follows: 1 C C S B = (µ i µ j ) T (µ i µ j ) (7) C(C 1) S W = 1 CM tr i=1 j=1 C i=1 k C i (H(:, k) µ i ) T (H(:, k) µ i ) Here C i meas the i th perso s images, µ i = 1 M tr H(:, k) deotes the mea vector of the i th class k C i i H. I the ext sectio, we shall derive the update rules for our model. Note that a similar idea has bee discussed ad the relevat model developed i [17]. But their fial update rule is etirely differet from ours. I [18], the authors idea is also similar to our work, however, our model uses just oe Lagragia multiplier α i the ew objective fuctio which is much more coveiet for parameter selectio i practical experimets. Moreover, (8) we used a differet defiitio for S W ad S B based o ubiased estimatio, ad the fial update rules are also differet from theirs. Furthermore, we shall also prove that usig our update rules, the o-egativity costraits for all the coefficiets i W ad H ca be satisfied automatically uder a mild coditio, ad this issue of maitaiig o-egativity was ot addressed i [18]. B. Modified update rules To derive our modified update rules, we will use a techique which miimizes a objective fuctio by usig a auxiliary fuctio similar to that used i [13]. Defiitio 1: G(H, H ) is a auxiliary fuctio for the fuctio F (H) if G(H, H ) F (H), G(H, H) = F (H) The we ca use the followig lemma: Lemma 1: If G is a auxiliary fuctio, the F is oicreasig by usig this update rule: H t+1 = arg mi H G(H, Ht ) where H t ad H t+1 deote the obtaied weight matrices after t iteratios ad t + 1 iteratios respectively. Proof 1: Sice F (H t+1 ) G(H t+1, H t ), G(H t+1, H t ) G(H t, H t ) ad G(H t, H t ) = F (H t ) we ca coclude F (H t+1 ) F (H t ). Usig this lemma, we ca obtai the correspodig update rule for the objective fuctio F 1. Notig that the ew terms i fuctio F 1 are just related to the weight matrix H, we ca directly adopt the update rule for W i the origial NMF algorithm, the just deduce the iterative algorithm for H t+1 whe fixig W ad give H t. First, we ca desig the followig auxiliary fuctio for F 1 : Lemma 2: Defie G 1 (H, H t ) = i,j i,j,k (V ij log V ij V ij ) + i,j V ij (W H) ij W ik Hkj t W i Hj t (log(w ik H kj ) W ikh t kj log W i Hj t ) + αs W αs B It s a auxiliary fuctio for F 1 (H). Proof 2: Proof G 1 (H, H t ) = F 1 (H) is straightforward to verify. To show that G 1 (H, H t ) F 1 (H), we use the covexity of the log fuctio to get the followig iequality: log( k Settig α ijk =, based o the above iequality we ca coclude W ik H kj ) k if α ijk 0 ad k W ikh t kj W ik Hkj t k log(w H) ij k α ijk log W ikh kj α ijk, α ijk = 1 W ik Hkj t W i Hj t (log W ik H kj

5 1298 JOURNAL OF SOFTWARE, VOL. 9, NO. 5, MAY 2014 W ikh t kj log W i Hj t ), i.e. G 1 (H, H t ) F 1 (H) Based o this lemma, we miimize F 1 (H) by usig the followig rule H t+1 = arg mi H G 1(H, H t ) The miimum of G 1 (H, H t ) is obtaied by settig G 1 (H,H t ) H βγ = 0 for ay β, γ. Let H(:, γ) be the weight vector of a image belogig to the i 1th class. Sice G 1 (H, H t ) H βγ = W iβ Hβγ t 1 V iγ W i i Hγ t + H βγ i + 2αH 2α H βγ1 βγ γ 1 C i1 CM tr 4α M tr C(C 1) CM 2 tr we obtai a quadratic equatio for H βγ : where b = 1 6α ah 2 βγ + bh βγ + d = 0 CMtr 2 Hβ+ t C i1, γ a = 2αM tr 6α CMtr 2, d = Hβγ t W iβ j i 1 (µ i1 (β) µ j (β)), 4α µ M tr C(C 1) j (β) i j i1 iβ V iγ ( H t ) iγ The we derive the formula for the bigger of the two real roots, amely, H t+1 βγ : H t+1 βγ = b + b 2 4ad (9) 2a Usig the derivatio i [13], we ca obtai the update rules for W : W t+1 λβ = W λβ t V λγ (W t H t Hβγ t (10) ) λγ γ λβ = λβ Ŵ t+1 γ γβ (11) Comparig these fial update rules with the origial NMF rules (i.e. equatios (5) ad (9)), we see that the mai differece betwee our ew algorithm ad the traditioal NMF method is the ew update rule for H which itegrates the class iformatio i the weight vectors of the traiig set. C. Proof of No-egativity From the previous sectio, the update rules for W is the same as i the covetioal NMF ad its oegativity is esured [13]. However, the icorporatio of class iformatio via LDA has yielded very differet update rule for the weight matrix H. I this sectio, we shall establish a sufficiet coditio to esure its oegativity. We shall proceed by iductio ad assume that H t > 0. The we have We also ote that H t, > 0 d < 0 M tr > 3 a > 0 the sice d < 0, from (9) we have M tr > 3 a > 0 b 2 4ad > b 2 H t+1 βγ > 0 Ad sice H 0 > 0, by iductio we have H t > 0, t. Therefore, we have established a sufficiet coditio for the o-egativity of H, amely M tr > 3, i.e. the umber of traiig images must be 4 or more. Based o the above aalysis, we coclude that our update rules will produce a sequece of oicreasig values of the ew objective fuctio, which is at least coverget to a locally optimal solutio ad satisfies the o-egativity costraits as log as sufficiet umber of traiig images are used. Lastly, we ote that although the coditio M tr > 3 is ot strictly ecessary for the o-egativity of H t+1 (sice H t+1 βγ > 0 wheever b < 0), i practice the sig of b is data-depedet, thus it s advisable to treat the coditio as if it is a ecessary coditio such that the update rules wo t break dow i ay case ad maitais oegativity. Sice most practical face databases geerally iclude may images for each perso, it s a mild coditio i real applicatios. IV. TESTING DATABASES USED IN THIS PAPER A moder face recogitio system [19], [20] has to deal with realistic situatios ad achieve good performace o challegig databases which should iclude cosiderable pose variatios ad real-life illumiatio chages etc. Therefore, we use the followig well kow face databases for the performace evaluatio of our method. A. ORL database The Olivetti database, as the ame suggests, origiated at the Olivetti Research Laboratory (ORL) i Eglad. I ORL database, there are 40 persos ad each perso cosists of 10 images with differet facial expressios ad illumiatio, small scale ad small rotatio. B. CMU PIE database I CMU Pose, Illumiatio, ad Expressio (PIE) database, there are 68 persos ad each perso has 13 pose variatios raged from right profile image to left profile image ad 43 differet lightig coditios, 21 flashes with ambiet light o or off. I our experimets, for each perso, we select 56 images icludig 13 poses with eutral expressio ad 43 differet lightig coditios i frotal view. For all frotal view images, we apply aligmet based o two eye ceter ad ose ceter poits but o aligmet is applied o the other images with pose.

6 JOURNAL OF SOFTWARE, VOL. 9, NO. 5, MAY C. FERET database The Facial Recogitio Techology (FERET) database was sposored by the Departmet of Defeses Couterdrug Techology Developmet Program [21]. We selected 120 persos, 6 frotal-view images for each idividual. Face image variatios i these 720 images iclude illumiatio, facial expressio, partial occlusio ad agig [21]. All images are aliged by the ceters of eyes ad mouth. To reduce the computatioal complexity, we ormalized all the images to the same resolutio, 23 28, by earest eighbour iterpolatio. We also ormalized the pixel values of each image i the above databases to [0, 1]. V. EXPERIMENTS I this sectio, we build a face recogitio system to provide a direct compariso of PCA, NMF, Fisherface, sequetial NMF+LDA ad our LDA-based NMF usig images from databases described i Sect.IV, which iclude complicated variatios i illumiatio, pose ad expressio. I all the experimets, we radomly selected M tr images per perso from the databases to form a traiig set ad use the remaider as the test set. The the system adopts the aforemetioed 5 approaches ad all of them cosist of two stages, amely, traiig ad recogitio stages. Traiig stage computes the represetatioal bases for traiig images ad coverts them ito traiig image represetatios, the all the represetatios are stored ito the library. Recogitio stage traslates the probe image ito probe image represetatio usig the represetatioal bases, ad the, matches it with those traiig images stored i the library to idetify the face image. The detailed procedure for PCA, Fisherface ad NMF ca be foud i refereces [6], [9], [16], so we just metio the two stages for our LDA-based NMF. A. Traiig stage There are 3 major steps i the traiig stage: First, we use a m matrix V 1 to represet all the traiig images i oe database. I the secod step, LDA-based NMF algorithm is applied to V 1 ad we ca fid two ew matrices (W 1 ad H 1 ) s.t. r (V 1 ) ij (W 1 H 1 ) ij = (W 1 ) ia (H 1 ) aj a=1 as described i the sectio 3 to obtai the base matrix W 1. Fially, let W + = (W1 T W 1 ) 1 W1 T, the each traiig face image V t is projected oto the liear space as a feature vector H t = W + V t which is the used as a prototype feature poit. B. Recogitio stage The recogitio stage ca be divided ito 2 steps. 1) Feature extractio: Each probe image V p to be classified is represeted as H p = W + V p. The, we will get the weight matrix H 2 for the probe set. 2) Nearest eighbor classificatio: I this step, some suitable distace betwee the probe image ad traiig image, dist(h p, H t ), is calculated, the the probe image is classified to the class which the closest traiig image belogs to. C. Distace measures ad parameter selectio The Mahalaobis distace is oe of the most widelyused distace measures i patter recogitio tasks [15], [22], so it s applied as a replacemet for the Euclidea distace i this classificatio task. The defiitio of Mahalaobis distace is as below: d(x, Y ) = (X Y ) σ 1 (X Y ) (12) Where X, Y are feature vectors of legth obtaied by differet methods, ad σ is the auto-covariace matrix for weight vector of traiig images. I our experimets, we use the full covariace matrix (as opposed to the simpler diagoal covariace matrix) sice it always obtais the better result for our face databases. The regularizatio parameter α i the ew cost fuctio (6) is importat as it determies the amout of weight to be assiged to the discrimiat factor. Accordig to our experiece, the best results are geerally obtaied whe α is about 0.1. D. Results from experimets A set of experimets were coducted o the above system, the we evaluated the classificatio performace of LDA-based NMF algorithm with the Mahalaobis distace metrics ad compared it with the result of PCA, NMF, Fisherface ad the sequetial NMF+LDA algorithm. I all the experimets, M tr images per perso were selected from the database to form a traiig set ad the rest are used as the test set. The Nearest Neighbor Classificatio method is adopted to obtai the correspodig recogitio rate. Followig [23], to produce reliable results, we take 10 repetitios of the radom split for each database ad the obtai the results i terms of the average recogitio rate (over the 10 differet splits) as well as the stadard deviatio of the recogitio rates, which we icorporate i the result plots i Fig.3, Fig.4 ad Fig.5 as error bars idicatig plus or mius oe stadard deviatio from the mea.

7 1300 JOURNAL OF SOFTWARE, VOL. 9, NO. 5, MAY 2014 Recogitio rate ORL database: M tr = Dimesioality of the weight vectors NMF NMF+LDA Eigeface Fisherface Our method Figure 3. Average recogitio rates accordig to the 5 methods for ORL database whe M tr = 6. Recogitio rate CMU PIE database: M tr = Dimesioality of the weight vectors NMF NMF+LDA Eigeface Fisherface Our method Figure 4. Average recogitio rates accordig to the 5 methods for CMU PIE database whe M tr = 5. Recogitio rate FERET database: M tr =4 NMF NMF+LDA Eigeface Fisherface Our method Dimesioality of the weight vectors Figure 5. Average recogitio rates accordig to the 5 methods for FERET database whe M tr = 4. From these figures, we ca see that the NMF class of methods ad the Fisherface performs much better tha the PCA approach. We believe that the superior performace may be attributed to either the use of LDA (i whatever form) or the advatage of the parts-based represetatio of the NMF-based methods. For the ORL database, our method is superior to the traditioal NMF, sequetial NMF+LDA ad Eigeface, ad is comparable to the Fisherface method. For the CMU PIE database, which cotais more complicated pose variatios ad real-life illumiatio chages, our method is sigificatly superior to all the other methods whe the dimesioality of feature vectors are ot very big. But the differeces become egligible as the dimesioality icreases ad our method is practically the same as the Fisherface ad the NMF+LDA, but still much better tha the traditioal NMF ad the Eigeface. For the FERET database, oce agai the three LDAbased methods outperform the traditioal NMF which i turs outperforms the Eigeface method. The advatage of our method over Fisherface is still evidet (albeit to a lesser extet) while the sequetial NMF+LDA gives very comparable results to our approach except at large dimesioality whe our method is margially better. To summarize, our LDA-based NMF algorithm is much better tha the traditioal NMF ad Eigeface, ad i specific cases, is at least comparable to the Fisherface method ad the sequetial NMF+LDA method which both use the same class iformatio i the traiig set. However, our method is the oly method that cosistetly achieves the best results for all the experimetal databases used. VI. CONCLUSIONS & FUTURE WORK I this paper, we proposed a ew costraied oegative matrix factorizatio algorithm, called LDA-based NMF, for the face recogitio problem. Its basic idea is to add the Fisher discrimiat to the cost fuctio of the oegative matrix factorizatio model. We showed that it ca perform better tha PCA ad the origial NMF for face recogitio. Furthermore, we foud its performace ca be better tha Fisherface ad the sequetial NM- F+LDA method which also take advatage of the same class iformatio i the traiig images. However, the improvemet depeds o the face database used ad the dimesioality of the feature vectors. This suggests that our method ca be improved if used i cojuctio with a feature selectio algorithm. Recetly, several differet NMF algorithms [17], [24] have bee proposed, ad the advatages ad disadvatages of them are compared based o differet aspects. But for the task of face recogitio, we still eed further work to ehace their performace. Based o the fidigs i this paper, we believe that the itroductio of suitable additioal costrait ca lead to more robust base vectors ad improve the fial recogitio result. As with the PCA method, NMF also suffers from the problem of large computatioal load. To resolve this limitatio, we are cosiderig the use of image trasform

8 JOURNAL OF SOFTWARE, VOL. 9, NO. 5, MAY to reduce the computatioal complexity ad give better recogitio accuracy. I particular, wavelet trasform has bee a very popular tool for image aalysis, ad we are cosiderig applyig our modified NMF method to selected subbads i our future work. ACKNOWLEDGEMENTS This research was partially supported by the Natioal Natural Sciece Foudatio of Chia (Grat Nos ) ad Sciece ad Techology Plaig Project of Guagdog Provice of Chia uder Grat 2009B , 2009B REFERENCES [1] R. Chellappa, C. L. Wilso, ad S. Sirohey, Huma ad machie recogitio of faces: a survey, Proceedigs of the IEEE, vol. 83, o. 5, pp , [2] G. Chow ad X. Li, Towards a system for automatic facial feature detectio, Patter Recogitio, vol. 26, o. 12, pp , [3] F. Goudail, E. Lage, T. Iwamoto, K. Kyuma, ad N. Otsu, Face recogitio system usig local autocorrelatios ad multiscale itegratio, IEEE Tras. PAMI, vol. 18, o. 10, pp , [4] K. M. Lam ad H. Ya, Locatig ad extractig the eye i huma face images, Patter Recogitio, vol. 29, o. 5, pp , [5] M. Kirby ad L. Sirovich, Applicatio of the karhueloeve procedure for the characterizatio of huma faces, IEEE Tras. PAMI., vol. 12, pp , [6] M. Turk ad A. Petlad, Eigefaces for recogitio, J. Cogitive Neurosciece, vol. 3, pp , [7] D. D. Lee ad H. S. Seug, Learig the parts of objects by o-egative matrix factorizatio, Nature, vol. 401, pp , [8] D. Guillamet ad J. Vitrià, Classifyig faces with oegative matrix factorizatio, i Proceedigs of the 5th Catala Coferece for Artificial Itelligece(CCIA 2002). Castello de la Plaa,Spai: Spriger-Verlag, 2002, pp [9], No-egative matrix factorizatio for face recogitio, i CCIA 02: Proceedigs of the 5th Cataloia Coferece o AI. Lodo, UK: Spriger-Verlag, 2002, pp [10] Y. Xue, C. S. Tog, Y. Che, ad W. S. Che, Clusterigbased iitializatio for o-egative matrix factorizatio. Applied Mathematics ad Computatio, vol. 205, o. 2, pp , [11] Y. Xue, C. S. Tog, W. S. Che, W. P. Zhag, ad Z. Y. He, A modified o-egative matrix factorizatio algorithm for face recogitio. i ICPR (3). IEEE Computer Society, pp [12] A. K. Jai, Fudametals of digital image processig. Pretice Hall, 1989, pp [13] D. D. Lee ad H. S. Seug, Algorithms for o-egative matrix factorizatio, Adv. Neural Ifo. Proc. Syst., vol. 13, pp , [14] Y. Xue ad C. S. Tog, Evaluatio of distace measures for mf-based face recogitio, i Proceedigs of the 2006 Iteratioal Coferece o Computatioal Itelligece ad Security Part I, Y. mig Cheug, Y. Wag, ad H. Liu, Eds., Nov. 2006, pp [15] R. Duda, P. Hart, ad D. Stork, Patter Classificatio. Wiley, [16] P. N. Belhumeur, J. Hespaha, ad D. J. Kriegma, Eigefaces vs. fisherfaces: Recogitio usig class specific liear projectio, IEEE Tras. PAMI., vol. 19, o. 7, pp , [17] Y. Wag, Y. Jia, C. Hu, ad M. Turk, Fisher o-egative matrix factorizatio for learig local features, Asia Coferece o Computer Visio, Korea, Jauary 27-30, [18] S. Zafeiriou, A. Tefas, I. Buciu, ad I. Pitas, Exploitig discrimiat iformatio i oegative matrix factorizatio with applicatio to frotal face verificatio, IEEE Tras. o Neural Networks, vol. 17, o. 3, pp , [19] Fabie Cardiaux, Corad Saderso, ad Samy Begio, User Autheticatio via Adapted Statistical Models of Face Images, IEEE Trasactio o Sigal Processig, vol. 54, o. 1, pp , [20] C. Saderso, S. Begio, ad Y. Gao, O trasformig statistical models for o-frotal face verificatio, Patter Recogitio, vol. 39, o. 2, pp , [21] P. J. Phillips, H. Moo, S. A. Rizvi, ad P. J. Rauss, The FERET evaluatio methodology for face-recogitio algorithms, IEEE Tras. PAMI., vol. 22, o. 10, pp , [22] C. M. Bishop, Patter Recogitio ad Machie Learig (Iformatio Sciece ad Statistics), 1st ed. Spriger, [23] P. S. Aleksic ad A. K. Katsaggelos, Audio-visual biometrics, IEEE Proceedigs, vol. 94, pp , November [24] W. Liu ad J. Yi, Existig ad ew algorithms for oegative matrix factorizatio, Tech. Rep. Yu Xue received his Ph.D. degree i image processig from Hog Kog Baptist Uiversity, Hog Kog, i He is curretly a associate professor i South Chia Normal Uiversity ad a member of the IEEE ad ACM. He has published more tha twety papers. His curret research iterests iclude patter recogitio ad data miig.