General Regression and Representation Model for Face Recognition

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1 013 IEEE Conference on Computer Vson and Pattern Recognton Workshops General Regresson and Representaton Model for Face Recognton Janjun Qan, Jan Yang School of Computer Scence and Engneerng Nanjng Unversty of Scence and Technology, Nanjng, Chna Abstract Recently, the regularzed codng-based classfcaton method (e.g. SRC and CRC) shows a great potental for face recognton. However, most estng codng methods gnore the statstcal nformaton from the tranng data, whch actually plays an mportant role n classfcaton. To address ths problem, we develop a general regresson and representaton model (GRR) for classfcaton. GRR not only has advantages of CRC, but also ntroduces the pror nformaton and the specfc nformaton to enhance the classfcaton performance. In GRR, we combne the leave-one-out strategy wth K Nearest Neghbors to learn the pror nformaton from the tranng data. Meanwhle, the specfc nformaton s obtaned by usng the teratve algorthm to update the feature weghts of the test sample. Fnally, we classfy the test sample based on the reconstructon error of each class. The proposed model s evaluated on publc face mage databases. And the epermental results demonstrate the advantages of GRR over state-of-the-art methods. 1. Introducton Lnear regresson has been wdely appled to pattern classfcaton. To prevent overfttng, the L -regularzer s generally used n the lnear regresson model. In the past years, the L 1 -regularzer, whch s closely lnked to sparse representaton, becomes a hot theme n nformaton theory, sgnal/mage processng and related areas. Meanwhle, numerous fndngs of neuroscence and bology form a physologcal base for sparse representaton [1-3]. Recently, many efforts have been made to apply sparse representaton methods to pattern classfcaton tasks, ncludng sgnal/mage classfcaton and face recognton etc. Labusch et al. presented a smple sparse-codng strategy for dgt recognton and acheved state-of-the-art results on the MNIST benchmark [4]. J.C. Yang et al. addressed the problem of generatng a super-resoluton (SR) mage from a sngle low-resoluton nput mage va sparse representaton [5]. J. Maral et al. elaborated a framework for learnng mult-scale sparse representatons of mages wth applcatons to mage denosng and npantng [6]. J.C. Yang et al. employed sparse codng nstead of vector quantzaton to capture the sgnfcant propertes of local mage descrptors for mage classfcaton [7]. Partcularly, Wrght et al. ntroduced a sparse representaton based classfcaton (SRC) and successfully appled t to dentfy human faces wth varyng llumnaton changes, occluson and real dsguse [8]. A test sample mage s coded as a sparse lnear combnaton of the tranng mages, and then the classfcaton s acheved by dentfyng whch class yelds the least resdual. Subsequently, M. Yang and L. Zhang constructed a Gabor occluson dctonary for SRC to reduce the computaton cost by usng Gabor feature [9]. Although the newly-emergng SRC shows great potental for pattern classfcaton, t lacks theoretcal justfcaton. J. Yang et al. provded an nsght nto SRC and analyzed the role of L 1 -optmzer [10]. They thought that L 1 -optmzer contans two propertes sparsty and closeness. However, L 0 -optmzer can only acheve the sparsty. Sparsty determnes a small number of nonzero representaton coeffcents and closeness makes the nonzero representaton coeffcents concentrate on the tranng samples wth the same class label as the gven test sample. Wrght et al. gve a recent revew of sparse representaton for computer vson and pattern recognton [11]. In addton, many related tasks have been reported [1-16]. Wth the wdely use of sparse representaton for classfcaton, some scholars queston the role of sparseness for mage classfcaton [17, 18]. L. Zhang et al. analyzed the workng prncple of SRC and beleved that t s the collaboratve representaton that mproves the mage classfcaton accuracy rather than the L 1 -norm sparsty. Consequently, L. Zhang et al. presented a collaboratve representaton based classfcaton wth regularzed least square (CRC) [19]. Compared wth SRC, CRC delvers very compettve classfcaton results wth lttle computaton tme. Subsequently, M. Yang et al. proposed a relaed collaboratve representaton model (RCR) whch effectvely captures the smlarty and dstnctveness of dfferent features for pattern classfcaton [0]. Most of the prevous works focus on nvestgatng the mportance of the L 1 -regularzer/l -regularzer. However, /13 $ IEEE DOI /CVPRW

2 these works gnore some statstcal nformaton hdden n the tranng data. Ths paper ams to eplore the pror nformaton (learned from the tranng set offlne) and the specfc nformaton (learnng from the testng sample onlne) so as to enhance the classfcaton performance under dfferent condtons. To ths end, we propose a model named General Regresson and Representaton (GRR) for pattern classfcaton. The overvew of GRR s shown n Fg. 1. Compared to other classfcaton method, the novelty of the proposed model s twofold: Frst, GRR captures the pror nformaton from the tranng set va the generalzed Tkhonov regularzaton n conjuncton wth the leave-one-out strategy and K Nearest Neghbor method; Second, we propose two models of GRR: Basc GRR (B-GRR) and Robust GRR (R-GRR) by combnng the pror nformaton and the specfc nformaton wth dfferent strateges; To evaluate the proposed method, we perform eperments on the AR and the Etended Yale B databases, and the epermental results demonstrate the effectveness and robustness of the proposed GRR.. General Regresson and Representaton Model for Classfcaton (GRR) In ths study, GRR contans two models: Basc GRR and Robust GRR. We wll ntroduce these two models n detal as follows..1. Basc GRR Most of current works don t make full use of the statstcal nformaton of tranng set. To address ths problem, we ntroduce the concept of pror nformaton and propose a basc general regresson and representaton model for classfcaton. Specfcally, let A be the matr formed by the K nearest neghbors of the test sample from tranng set and y be the test sample. P s the weght matr of reconstructon errors, and Q s the weght matr of representaton coeffcents. P and Q are matrces contanng the pror nformaton. Our model s P Q ˆ arg mn y - A (1) We call the above model as the basc general regresson and representaton (B-GRR). Actually, ths model can be reformulated as follows: T T ˆ arg mn(y- A) P(y- A) Q () If P and Q are known, from [6], we know there s a close-form soluton: ˆ ( + T -1 T A PA Q) A Py (3) Fgure 1: An overvew of General Regresson and Representaton model for classfcaton However, P and Q are unknown beforehand. The remanng problem s how to learn P and Q. Here, we employ a generatve method to evaluate these two matrces. Specfcally, we use the leave-one-out strategy n conjuncton wth K Nearest Neghbor to learn the pror nformaton matrces P and Q. Denote by y the -th sample of tranng set. A s the K nearest neghbors of y from the tranng set. We can ntalze P 0, Q 0 as P 0 =I and Q 0 =I. The y s coded on A as follows: P 0 Q 0 ˆ arg mn y - A (4) Then, P and Q can be drectly derved by usng Eq. (5) and Eq. (6) respectvely: 1 P={ cov( E ) + 1I} (5) where E=[ e1,, e, ek] and e ˆ y A. Here, 1 s the regular parameter, and cov( ) s an operator to compute the covarance matr. 1 Q={ cov( X ) + I} (6) where X=[ ˆ ˆ ˆ 1,,, K ]. s the regular parameter. In the testng stage, the soluton of ˆ n Eq. (1) s easly derved by usng Eq. (3). We can reconstruct the test sample y as yˆ ( ˆ A ) by employng the coeffcents assocated wth -th class. The correspondng reconstructon error of -th class s defned: r( y) y- A ( ˆ) ( ˆ) (7) The decson rule s: f r( y) mnr( y ), y s assgned to Class l. l 167

3 Algorthm 1 B-GRR Input: Dctonary A, test sample y. Intal values P 0 and Q 0 1. Normalze the columns of A to have unt L -norm.. The pror nformaton matrces P and Q are learned from tranng set by usng the generalzed Tkhonov regularzaton, leave-one-out strategy and KNN. 3. The test sample y s coded on ts K nearest neghbors A va Eq. (1). 4. Compute the resduals of each class. Output: y s assgned to the class whch yelds the mnmum resdual. B-GRR makes full use of the pror nformaton of the tranng set. It works well when the testng samples share the same probablty dstrbuton wth the tranng samples. The algorthm of B-GRR for classfcaton s summarzed n Algorthm 1... Robust GRR In face recognton problems, llumnaton, epresson or pose changes may cause dfferences between test samples and tranng samples. Therefore, t s necessary to ntroduce the specfc nformaton of the test sample to allevate the effect caused by the dfferences between test samples and tranng samples. Ths specfc nformaton s to gve a weght to each mage pel, whch can be learned onlne va the teratvely reweghted method, as adopted n RSC [1]. Based on ths dea, we present a robust general regresson and representaton model (R-GRR) for classfcaton. Compared wth B-GRR, R-GRR not only ncludes the pror nformaton matrces P and Q, but also owns the specfc nformaton matr W. The model s gven below: ˆ arg mn 1/ - W (y A) (8) Q If P, Q and W are known, the above model has a close-form soluton: ˆ [ + T 1/ T 1/ -1 T 1/ T 1/ A (W ) P(W )A Q] A (W ) P(W )y (9) Snce P and Q can be learned offlne usng the same method as n Basc GRR, the remanng problem s to learn the specfc nformaton (matr W) onlne. Specfcally, gven a test sample y, we frstly compute the codng resduals e of y so as to ntalze the weght. The resdual e s ntalzed as e=y-y n, and y n s the ntal estmaton of the true mages from the observe samples. In ths study, we smply set y n as the mean mage of all samples n the codng dctonary snce we don t know whch class the test mage y belongs to. Wth the ntalzed y n, our method can estmate the W teratvely. W actually s a dagonal matr, W, P Algorthm R-GRR Input: Dctonary A, test sample y. Intal values P 0, Q 0 and y n. 1. Normalze the columns of A to have unt L -norm, test sample y wth L -norm and y t ntalzed as y n.. The pror nformaton matrces P and Q are learned from the tranng set by usng the generalzed Tkhonov regularzaton, leave-one-out strategy and KNN. 3. The test sample y s coded on ts K nearest neghbors A. () t () t a) Compute resdual e yy b)estmate weghts () t () t () t () t ep( ( e ) ) () t ( e ) () t () t () t () t 1 ep( ( e ) ) c) Code ˆ arg mn ( () t 1/ ) - (.e. ( e )) s the weght assgned to each pel of the test mage. The weght functon [1] s: ep( ( e ) ) e ( e ) (10) 1 ep( ( ) ) where, and are postve scalars. In addton, Eq. (9) s the eplct soluton of Eq. (8). The convergence s acheved when the dfference of the weghts between adjacent teratons satsfes the followng condton: W W W (11) () t ( t1) / ( t1) where, s a small postve value. The R-GRR algorthm for classfcaton s summarzed n Algorthm. 3. Further Analyss of GRR In ths secton, we wll further analyss the role of P, Q and W n GRR. P (a symmetrc matr) s the weght matr of reconstructon errors and learned from the tranng set. The dagonal elements of matr P stand for the mportance of mage pels or features. The non-dagonal elements represent the cross relatonshp between mage pels or P Q W (y A) d)compute the reconstructed test sample () t () t y A, and let t = t + 1 e) Go back to step a) untl the mamal number of teratons s reached, or convergence s met as shown n Eq. (11) 4. Compute the resduals of each class. Output: y s assgned to the class whch yelds the mnmum resdual. 168

4 Fgure : Two classes of samples from the AR database Test mage R-GRR RSC B-GRR CRC (a) Recovered clean mage and occluded part va four methods for the mage wth sunglasses Test mage R-GRR RSC B-GRR CRC (b) Recovered clean mage and occluded part va four methods for the mage wth scarf Fgure 3: The advantages of R-GRR features. The matr Q can be consdered as a regularzaton term. In general, the regularzaton term s determned manually. Here, Q s learned from the tranng set and represents the relatonshp between samples n the codng dctonary. We also gve an eample to show the advantages of Robust GRR (R-GRR) as shown n Fg. 3. In our eample, two classes of face mages from the AR database, as shown n Fg., are used for tranng. We test two cases of real-world dsguse mages: the mages wth sunglasses and the mages wth scarves. In Fg. 3 (a) and Fg. 3 (b), the left column contans the dsguse mages. In our test, we use R-GRR, RSC, B-GRR and CRC to deal wth occluson. For each occluded mage, the reconstructed mages (recovered clean mage) and the resdual mages (recovered occluson) are shown n Fg. 3. From Fg. 3, we can see that R-GRR acheves comparable result wth RSC and sgnfcantly outperforms other methods. 4. Epermental Results In ths secton, we perform eperments on publc face mage databases and compare the proposed model GRR wth state-of-the-art methods. Note that here n SRC and RSC, the matlab functon l1-ls [1] s used to calculate the sparse representaton coeffcents. In the followng eperment, the parameters and of Eq. (10) are determned by the rule n paper [1]. 1 and are set to 0.1, 10-7, respectvely. The parameter K s determned by eperments. Fg. 7 plots the recognton rates versus the varaton of the parameter K on the dfferent eperments. Dm NN LRC SRC CRC[19] RSC[1] B-GRR R-GRR Table 1. The recognton rates (%) of each classfer for face recognton on the AR database Dm NN LRC SRC CRC RSC B-GRR R-GRR Table. The recognton rates (%) of each classfer for face recognton on the Etended Yale B database 4.1. Face Recognton wthout Occluson We evaluate the performance of B-GRR and R-GRR on the AR and the Etended Yale B database wth llumnaton and epresson changes but wthout occluson. In these eperments, PCA s frst used to reduce the dmensonalty of face mage. AR Database The AR face database [] contans over 4000 color face mages of 16 persons, ncludng frontal vews of faces wth dfferent facal epresson, lghtng condtons and occlusons. The pctures of 10 ndvduals were taken n two sessons (separated by two weeks) and each sesson contans 13 color mages. Fourteen face mages (each sesson contans 7) of 100 ndvduals are selected and used n our eperment. The face porton of each mage s manually cropped and then normalzed to pels. In ths eperment, mages from the frst sesson are used for tranng, and mages from the second sesson are used for testng. Then LRC (lnear regresson classfcaton [14]), SRC, CRC, RSC, B-GRR and R-GRR are employed for classfcaton. The NN classfer s also used to provde a 169

5 Tranng mages Testng mages Tranng mages (a) Testng mages the proposed R-GRR acheves the best recognton rates n all dmensons for face recognton. When the feature dmenson s 100, R-GRR gves about 3% mprovement of recognton rate over LRC, SRC and CRC, respectvely. Methods Sunglasses Scarves CRC SRC[8] GSRC[9] CESR[15] RSC[1] R-GRR Table 3. The recognton rates (%) of each classfer for face recognton on AR database wth dsguse occluson (b) Fgure 4: Sample mages for one person of AR database. (a) Sample mages of the frst eperment. (b) Sample mages of the Second eperment. baselne. The parameter K of R-GRR means we choose the K nearest neghbors of test mage from tranng set to form the codng dctonary. K s set to 650 here. The recognton rates of each classfer versus the varaton of dmensons are lsted n Table 1. From Table 1, we can see that R-GRR gves better performance than state-of-the-art methods n all dmensons ecept that R-GRR s slghtly worse than RSC when dmenson s 54. However, t s dffcult to acheve better performance when dmenson s low for all the methods. The mamal recognton rates of NN, LRC, SRC, CRC, RSC, B-GRR and R-GRR are acheved when the dmenson s 300. Etended Yale B Database The etended Yale B face mage database [3] contans 38 human subjects under 9 poses and 64 llumnaton condtons. The 64 mages of a subject n a partcular pose are acqured at camera frame rate of 30 frames / second, so there are only small changes n head pose and facal epresson for those 64 mages. All frontal-face mages marked wth P00 are used n our eperment, and each s reszed to 48 4 pels. In our eperment, we use the frst 3 mages of each ndvdual for tranng and the remanng mages are used for testng. Based on the PCA-transformed features, NN, LRC, SRC, CRC, RSC, B-GRR and R-GRR are employed for classfcaton. The parameter K s 800. The recognton rates of each classfer correspondng to the varaton of feature dmensons are lsted n Table. Table shows that 4.. Face Recognton wth Occluson In ths secton, we eamne the robustness of R-GRR when face mages suffer dfferent occlusons, such as real dsguse or block occluson. Here, R-GRR combnes advantages of the pror nformaton Q and the specfc nformaton W to enhance the classfcaton performance and set P as unt matr. As we know, P reflects the probablty dstrbuton of reconstructon errors. If there are great dfferences between the test sample and the tranng samples, the resultng reconstructon error does not follow the orgnal dstrbuton. In ths case, usng P may cause negatve effect on the classfcaton performance. In the followng eperments, we manly compared our methods wth CRC, SRC, RSC, correntropy-based sparse representaton (CESR) [15] and Gabor-SRC [9]. Face Recognton wth Real Dsguse A subset of AR face mage database s used n our eperment. The subset contans 100 ndvduals, 50 males and 50 females. All the ndvduals have two sesson mages and each sesson contans 13 mages. The face porton of each mage s manually cropped and then normalzed to 4 30 pels. In the frst eperment, we choose the frst four mages (wth varous facal epressons) from the sesson 1 and sesson of each ndvdual to form the tranng set. The total tranng mages s 800. There are two mage sets (wth sunglasses and scarves) for testng. Each set contans 00 mages (one mage per sesson of each ndvdual wth neutral epresson). The sample mages of one person as shown n Fg. 4 (a). The parameter K s 300 for the test set wth sunglasses and 760 for the test set wth scarves. The face recognton results of each method on the two testng set are lsted n Table 3. From Table 3, we can see that R-GRR acheves the best recognton results among all the methods. Moreover, the performances of RSC and CESR 170

6 (a) (b) (c) (d) Fgure 5: The recognton rates (%) of each classfer for face recognton on AR database wth dsguse occluson. (a) The testng mages wth sunglasses from sesson 1. (b) The testng mages wth scarves from sesson 1. (c) The testng mages wth sunglasses from sesson. (d) The testng mages wth scarves from sesson. are both hgher when facal mage wth sunglasses. However, CESR only acheves 4% when facal mages wth scarves. In the second eperment, four neutral mages wth dfferent llumnaton from the frst sesson of each ndvdual are used for tranng. The dsguse mages wth varous llumnaton and glasses or scarves per ndvdual n sesson 1 and sesson for testng. The sample mages of one person as shown n Fg. 4 (b). We set the parameter K as 0, 300, 40 and 30 for the four test set, respectvely. The recognton results of each method are shown n Fg. 5. From Fg. 5, we can see clearly that R-GRR gves better performance than CRC, SRC, GSRC, CESR and RSC on dfferent testng subsets. Both SRC and CESR do well on the subsets wth sunglasses but poor n the cases wth scarves. However, GSRC acheves better result on the subsets wth scarves and worse result on the subsets wth sunglasses. Compared to RSC, at least 4.3% mprovement s acheved by R-GRR for each testng set. Meanwhle, t s worth notcng that the recognton rate of R-GRR s 71.6%, 63.6% hgher than SRC and CESR on the testng mages wth scarves from sesson, and 44.3% hgher than GSRC on the testng mages wth sunglasses from sesson. Recognton rate SRC GSRC 60 CESR RSC R-GRR Occluson percent Fgure 6: The recognton rates (%) of SRC, GSRC, CESR, RSC and R-GRR under the occluson percentage from 0 to 50 Face Recognton wth Block Occluson In ths eperment, we use the same eperment settng as n [8, 1] to test the robustness of R-GRR. Subsets 1 and of Etended Yale B are used for tranng and subset 3 s used for testng. The face mages are reszed to The parameter K s 00. Fg. 6 shows recognton rates curve of SRC, GSRC, CESR, RSC and R-GRR versus the varous levels of occluson (from 0 percent to 50 percent). From Fg. 6, we can see that the proposed R-GRR overall outperforms SRC, GSRC, CESR and RSC. When the occluson percentage s 50%, GRR acheves the best recognton rate 91.9, compared to 65.3 for SRC, 87.4 for GSRC, 57.4 for CESR, and 87.6 for RSC. It s surprsng that the performance of CESR s very poor. Probably, t s not sut to deal wth ths block occluson case. 5. Conclusons Ths paper presents a general regresson and representaton (GRR) model for face recognton. In GRR, we learn the pror nformaton from the tranng set by combnng the leave-one-out strategy and KNN n the framework of generalzed Tkhonov regularzaton. Also, we learn the specfc nformaton from the test sample by usng the teratvely reweghted algorthm. Actually, we provde two models: B-GRR and R-GRR, whch combne the pror nformaton and the specfc nformaton usng dfferent strateges. Eperments on face datasets demonstrated that the valdty of our model and ts performance advantages over state-of-the-art classfcaton methods. References [1] W. E. Vnje, J. L. Gallant, Sparse Codng and Decorrelaton n Prmary Vsual Corte Durng Natural Vson, Scence, 000, Vol. 87. no. 5456, pp [] B. A. Olshausen, D. J. Feld, Sparse codng of sensory nputs, Current Opnon n Neurobology, 004, Vol. 14, No. 4., pp [3] T. Serre, Learnng a Dctonary of Shape-Components n Vsual Corte: Comparson wth Neurons, Humans and Machnes, PhD dssertaton, MIT, 006. [4] Zhou, H.; Haste, T.; and Tbshran, R Sparse prncple component analyss. Techncal Report, Statstcs Department, Stanford Unversty. 171

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