Vol. 6, No. 8, 05 Compettve Sparse Representaton Classfcaton for Face Recognton Yng Lu Chongqng Key Laboratory of Computatonal Intellgence Chongqng Unversty of Posts and elecommuncatons Chongqng, Chna Jan-Xun M Chongqng Key Laboratory of Computatonal Intellgence Chongqng Unversty of Posts and elecommuncatons Chongqng, Chna Cong L Chongqng Key Laboratory of Computatonal Intellgence Chongqng Unversty of Posts and elecommuncatons Chongqng, Chna Chao L Chongqng Key Laboratory of Computatonal Intellgence Chongqng Unversty of Posts and elecommuncatons Chongqng, Chna Abstract A method, named compettve sparse representaton classfcaton (CSRC), s proposed for face recognton n ths paper. CSRC ntroduces a lowest compettve deleton mechansm whch removes the lowest compettve sample based on the compettve ablty of tranng samples for representng a probe n multple rounds collaboratve lnear representaton. In other words, n each round of competng, whether a tranng sample s retaned or not n the next round depends on the ablty of representng the nput probe. Because of the number of tranng samples used for representng the probe decreases n CSRC, the codng vector s transformed nto a low dmensonal space comparng wth the ntal codng vector. hen the sparse representaton makes CSRC dscrmnatve for classfyng the probe. In addton, due to the fast algorthm, the FR system has less computatonal cost. o verfy the valdty of CSRC, we conduct a seres of experments on AR, Extended YB, and ORL databases respectvely. Keywords face recognton; collaboratve representaton sparse representaton; and compettve representaton I. INRODUCION Face Recognton (FR) has become to a hot research area for ts convenence n daly lfe. Recently, lnear representaton methods are very popular whch represent the probe wth tranng samples from gallery set. Collaboratve representaton (CR) method has acheved good performance for FR [-4],n whch a gven testng mage y can be represented by a tranng set A wth a codng vector x,.e. y=ax. he tranng set A ncludng all samples from all subjects s an over-complete dctonary. It s known that face mages from a specfc class le n a lnear subspace and a probe can be represented by mages whch have the same label as the probe. Comparng to the sngle-representaton method, collaboratve representaton has more ablty to compensate the pxels of probe. In order to make the codng vector more dscrmnatve, the sparse constrant was ntroduced n regular term. Inspred by compressve sensng, the nduced sparse constrant on codng vector uses l 0 -norm so that the representaton problem s formulzed as: mn x st.. y Ax () 0 Where. denotes the 0 l -norm, whch counts the 0 number of nonzero entres of the codng vector and s a small error tolerance. he l -norm s wdely dscussed and used n many 0 researches. However, the problem of fnd the sparsest soluton of an underdetermned system suffers the ssue of NP-hard. Researchers put forward dfferent solutons to l 0 -norm [5-7]. Now, the sparse constrant methods n CR can be dvded nto two categores: frst one uses l -norm constrant nstead of l 0 - norm, and second one employs supervsed sparse scheme. he frst CR method uses l -norm n FR s Sparse Representaton Classfcaton (SRC) []. In SRC, the sparse representaton s solved usng Lasso formulaton n whch the sparse degree of the codng vector can be adjusted by the norm constrant ntensty. However, the l -norm constrant could not brng good performance when the tranng samples have hgh correlaton [8]. In addton, l -norm based sparse representaton problem s qute tme consumng. A two-phase sparse representaton (PSR) was proposed by Xu. et.al usng supervsed sparse constrant scheme [9]. he M nearest neghbors of a probe are selected based on the frst phase of representaton and then used as the new tranng set n the second phase of representaton n PSR. However, the onetme deleton n PSR may lead to all samples from the classes as probe are removed when the probe s serously dstorted. In addton, PSR s senstve to llumnaton because of the samples wth negatve coeffcents are lkely removed. In ths paper, we propose a supervsed sparse constrant method named as compettve sparse representaton classfcaton (CSRC). Based on the compettve ablty of each tranng sample for representng a probe, the proposed CSRC ntroduces a lowest compettve deleton mechansm whch removes the lowest compettve samples based on the compettve ablty of each tranng sample n collaboratve P a g e
Vol. 6, No. 8, 05 lnear representaton. Only those samples wth hgh compettve can be used n the next collaboratve representaton. hen the dmensonalty of the codng vector n the next representaton s bgger than the current one. he mult-phase deleton s more useful for classfcaton than two-phases deleton [0]. Accordng to that the probe s represented based on the mult-phase deleton and untl the condton s satsfed. Meanwhle, the dmensonalty of the fnal codng vector s much smaller than the frst one s,.e. the codng vector s sparse. In CSRC, the compettve ablty of samples from correct class as probe s ncreasng as the lowest compettve ones are removed. In addton, the fast algorthm of CSRC enhances the effcency of the FR system, because the algorthm avods the procedure of fndng nverse matrxes n each collaboratve representaton. One advantage of CSRC s the mult-phases deleton lets the compettve ablty of the correct class s strengthened gradually and avods all samples form correct class of probe are removed n the one-tme collaboratve representaton. he other one s that comparng wth l -norm sparse constrant, CSRC has lower computatonal complexty wth the fast algorthm. hs paper s organzed as followed: n secton, three parts are descrbed: the ntroducton about a basc general framework for classfcaton usng compettve sparse representaton, descrpton the optmzaton method, and the analyss of the computatonal cost of CSRC. he features of CSRC are descrbed n secton 3.We conduct a seres of experments to verfy the good performance of our method n secton 4 and the conclusons are demonstrated n secton 5. II. COMPEIIVE SPARSE REPRESENAION CLASSIFICAION Face mages from a same class le n a lnear subspace, a query mage can be represented by wthn-class samples [, ]. But face recognton s a lack of samples problem n general. When the number of tranng samples of each class s not bg enough, t s hard to obtan a good representaton of a query mage by a small part of tranng samples that from a sngle-subject. hat s to say, the representaton has large dstance wth the query mage. hus, the recognton result s unstable. However, the query mage can be represented fathfully by collaboratve lnear representaton whch each tranng samples are put n the dctonary (sometmes the dctonary s over-completed). Because more tranng samples partcpate n representng the query mage n collaboratve lnear representaton, the compettve of tranng samples that from the class as the query decreases. As a consequence, the query mage s lkely classfed nto the wrong class. However, n the collaboratve representaton, not every sample has hgh compettveness (hgh coeffcent value). So removng these less compettve mages from the dctonary can ncrease the compettve of samples from correct class. he lowest compettve deleton mechansm n CSRC, whch delete the lowest compettve tranng samples from the dctonary n mult-phases. hs mechansm can ncrease the compettve of the correct samples through removng the lowest compettve samples. In the meantme, as the samples are removed n CSRC, the dmensonal of representaton coeffcents are smaller. Compared wth the space of the ntal representaton coeffcents, the fnal representaton coeffcents le n a subspace of t. In other words, the representaton coeffcents are sparse. he sparse codng vector (representaton coeffcents) has more dscrmnate nformaton. A. Compettve Sparse Representaton Classfcaton Gven suffcent tranng samples of the th object class, m m n A =a (,,..., c ), where n denotes the n, R R number of tranng samples and each class contans n tranng samples. Meanwhle each tranng sample s an m - m dmensonalty feature vector. A probe mage y R s represented by collaboratve lnear representaton over the dctonary. Snce CSRC s a mult-phase deleton method, the t t symbol A and x ( t,,... ) denote the tranng samples and codng vector n t th collaboratve representaton respectvely. he representaton framework s wrtten as followng: mn x s.t. y A x () Where s the nosy term. he Eq. () can be wrtten as rdge regresson form: x =arg mn y A x x (3) x he codng vector can be computed as: - n x =( A A + I ) A y x R (4) n n Where the unt matrx I R and s a Lagrangan coeffcent. Snce CSRC deletes only one tranng sample n each phase, CSRC removes the least compettve samples based on the correspondng entres of the codng vector,.e., t fnds the mnmum absolute value x amongst the representaton coeffcents xj mn{ x, x... x n }, and remove the correspondng tranng sample. So the dctonary s dvded nto two subsets: the frst subset ncludes the deleted mage, and the second subset ncludes the retaned samples. Here the samples n the second part wll be used as a r dctonary n the second phase. Let A and A denote the removed sample after the frst representaton and the tranng samples n th representaton respectvely. So the above two r m A a a and subsets can be descrbed as j j R r m (n ) A A A, A R j a j. hen the test y can be represented over the new dctonary A. In the same way, repeatedly conduct the above operaton n the next representaton phase. Assume the k th collaboratve representaton reaches the maxmum number of the deleton phases. he fnal codng k vector x can be represented as followng: k k k - k k n k x =( A A + I) A y x R (5) Snce many samples are removed from the dctonary A P a g e
Vol. 6, No. 8, 05 k that s used n the frst representaton, t s very lkely that A excludes all the samples of some classes. herefore, the orgn classfcaton problem s weakened to a smpler problem whch contans fewer classes. he coeffcents of the deleted samples as zero and then select the coeffcents assocated wth the th class and mark t as δ,,,..., c. hen the Eucldean dstance s used for measurng the dstance between each class and the test mage y. he rule of classfcaton s n favor of the class wth mnmum dstance. he formula can be expressed as followng: ID() arg mn d ( y) arg mn y - Aδ,,..., c (6) he detaled algorthm s gven as followng: Algorthm: Compettve Sparse Representaton Classfcaton (CSRC). Input: an undentfed mage n y R, the ntal dctonary A R nm.. Intal value: t 3. Repeat 4. Compute the codng vector accordng to (3) 5. Removng the tranng sample a j from the dctonary t A 6. t Update the dctonary : A, t t 7. Untl satsfy termnaton condtons 8. Identfy : ID arg mn( yaδ ) 9. Output: the dentty of y as. B. Optmzaton As t known to all, the analytcal soluton of the above lnear model s x ( A A I) A y. Due to the deleton operaton, the dctonary s updated n each phase. However, the new dctonary s the subset of the last dctonary. CSRC mplements a fast algorthm to avodng the repeated matrx nverson calculaton. Now let a new symbol A expresses the elementary transformaton of A,.e. A AE, where E s an elementary matrx. A can be treated that the matrx contans two matrces s A and r s r r s A A, A. he two matrces A and A A,.e. denote the deleted sample and the new tranng samples respectvely. Snce the matrx A s gven, so the nverse matrx ( A A I ) s avalable. hen the matrx ( A A I) can be derved from ( A A I ). he detaled dervaton processes are wrtten as follows: ( A A I) (( AE) ( AE) I) ( E A AE I) (7) ( E A AE E IE) ( E ( A A I) E) ( ( ) ) ( ) E A A I E E A A I E he key problem for solvng the codng vector s nverse matrx. In order to have a convenence expresson n (7), the equaton can be represented by the four matrces (O, P, C, and V ) as: ( A A ) s s ss s r sr I r s rs r r rr A A I A A I O P A A I A A I C V (8) Accordng the elementary transformaton of matrx, the nverse matrx wll be transformed as followng:. O P Is 0 O P Is 0 - - C V 0 Ir 0 V CO P CO Ir hen ( A A I) - - - - - A 0 Is + P( V CO P) CO - P( V CO P) = - - 0 V CO P CO Ir - - - - - O 0 Is + P( V CO P) CO - P( V CO P) = - - - - - 0 Ir ( V CO P) CO ( V CO P) - - - - - - - - Is 0 O + O P( V CO P) CO O P( V CO P) = - - - - - 0 Ir ( V CO P) CO ( V CO P) (9) - - - - - - - - O P O + O P( V CO P) CO - O P( V CO P) - - - - - C V ( V CO P) CO ( V CO P) (0) In fact, the ultmate goal s to obtan the soluton nverse s matrx about A, whch s used as the new dctonary n the next teraton, then the nverse matrx can be wrtten as s s s s A A I. Not hard to fnd that s s s s ( A A I ) O O + O P( V CO P) CO () - - - - - - Where the delghted thngs are that O, P, C and V are already known n the matrx A. A group of new symbols are ntroduced to express the four block of ( A A I ),.e., - - - - - Q O O P V CO P CO () + - - - Q O P V CO P (3) - - - - Q V CO P CO (4) - - - Q V CO P (5) Combnng () and (3), (4), and (5) respectvely, the four block matrces can be descrpted as followng Q O O PQ Q Q Q V CO P - - - -O PQ - -QCO - - (6) 3 P a g e
Vol. 6, No. 8, 05 After a few tmes terate replacements, the expresses by other blocks,.e., - - - O can be O Q Q Q Q (7) Here Q s a nvertble square matrx n general case. Snce only one the lowest tranng sample s removed n each phase, so Q s a scalar. In other words, the analytcal soluton about the nverse matrx s s s s A A I s obtaned,.e. It s easy to fnd that CSRC reduces the nterference of the wrong classes. s s s s ( A A I ) Q Q Q Q (8) - C. Complexty analyss We only analyze the tme complexty of (5) n ths secton, snce ths process s the most tme consumng n algorthm of CSRC. It s tme consumng way to solve (5) drectly n a certan collaboratve representaton and the tme complexty s 3 O( n m+n ) However, the tme complexty that CSRC obtans the codng vector s much less than (9). Snce the matrx nverson n (5) s replaced by (8). Moreover, the matrx Q n (8) s only one element, t save more calculaton. herefore, the complexty of CSRC s denoted as O( n m ). In addton, t s much less than the complexty that SRC obtans the spares.5 codng vector,.e. O( n m ). III. ANALYSIS OF CSRC he method Collaboratve representaton based on Classfcaton (CRC) uses l -norm constrant codng vector n the collaboratve representaton [3]. Although the codng vector s not sparse, t fully embodes the compettve level of each tranng sample n CRC. However, CRC obtans the regresson model uses only one collaboratve representaton. When the number of the tranng samples s small, t may lead to regresson model over-ftted. he compettve representaton, adopted n CSRC, removes the lowest compettve tranng sample from the tranng set n the current round and the rest tranng samples wll be used n the nest round. After several rounds of competng, all samples of the subject whch has a low correlaton wth the query may be removed. So CSRC reduces the scale of the FR problem. he Fg., depcts the resduals between the probe and predcton of each class whch are calculated by CRC and CSRC respectvely, llustrates ths phenomenon on ORL database. he upper one n the fgure s obtaned by CRC and the bottom one s obtaned by CSRC. he dstance between the probe and the predctons from over 0 classes s, whch means the tranng samples from these classes are removed and the coeffcents respect to them are zeros. In addton, n CSRC method, the lowest compettve deleton mechansm reduces resdual the correct class of the probe and enlarges the resduals of the wrong classes. Accordng to the fgure t s easy to calculate that the rato of two smallest resduals by CRC and CSRC are.579 and.77 respectvely. he rato of two smallest resduals s enlarged by CSRC, whch means CSRC has better dscrmnatve than CRC. Fg.. he resdual mages by CRC and CSRC for a clear testng face on the ORL database. We select the ffth mage of the frst man as a probe and the frst fve mages as tranng samples. In above two hstograms, the horzontal axs denotes the number of the class and the vertcal axs denotes the resduals between the probe and each class. he top one: the two smallest resdual are 0.49 and 0.7774 by CRC and the rato of them s.579. he bottom one: two smallest resdual are 0.4895 and 0.84.3 by CSRC and the rato of them s.77 IV. EXPERIMENAL RESULS o evaluate the proposed CSRC algorthm, we conduct a serous of experments on mages from AR database, Extended YB database and ORL database respectvely, as well as comparng wth state-of-the-art methods ncludng CRC, SRC (wthout extended matrx), and PSR (the canddate set s 0%). We also assess the recognton rate of CSRC(the canddate set s 0%) on the occluded testng faces. All experments are performed n MALAB on 04b on desktop wth 4GHZ CPU and 8G RAM. A. Face Recognton wthout Occluson AR database More than 4000 color face mages of 6 people (70 men and 56 women) consst n AR database [4]. Each people has 6 mages nclude frontal vews of face wth dfferent facal expresson, llumnaton and occluson. he pctures of each ndvdual were taken n two sessons (separated by two weeks). Each secton contans 3 color mages and 0 ndvduals (65 men and 55 women) partcpated n both sessons. he mages of these 00 ndvduals (50 women and 50 men) were selected and used n our experment. Faces that are used to test these methods are gray and then normalzed t to 50 40 pxels. 4 P a g e
Vol. 6, No. 8, 05 We select the frst seven faces n sesson one as tranng samples and the frst seven faces n the sesson two as the testng samples for each class and a specfc class faces are shown n Fg.. he recognton rates of the four methods are shown n the ab.. Snce the testng samples and tranng samples are collected n dfferent tme, none of all methods have a 00% recognton rate. Snce the tranng samples have hgh correlaton, the sparse representaton by SRC could not obtan a good performance. However, snce CRC could not ncrease the compettve ablty of the samples from the correct class as probe, so CRC has lower result than PSR and CSRC. Furthermore, PSR has lower recognton rate than CSRC, because of that PSR s lkely to delete all mages from the correct class n the frst phase. From the experment we can see that the lowest compettve deleton mechansm n CSRC makes the codng vector has more dscrmnant nformaton ndeed. Extended Yale B database he extended Yale B face database contans 38 persons under 64 llumnaton condtons [5,6]. A subset (contans 3 ndvduals) s used n ths experment. he 64 mages of a person n a partcular pose are acqured at camera frame rate of 30 frames/ second, so there s only small change n head pose and facal expresson for those 64 mages. Each mage s reszed to 50 40 pxels n our experment. Several frontal faces of one person are shown n Fg. 3. As s known to all, llumnaton s another bg challenge for face recognton. Faces were captured under carous laboratory-controlled lghtng condtons. Samples n subset one (seven mages per person) under nomnal lghtng condton was used as the gallery. Snce the recognton rate for test subset and 3 (characterze slght-to-moderate lumnance varatons) are by all methods. Here we select faces n subset 4 are used for verfy CSRC method. Due to the ncreasng llumnaton condton, the recognton results are not very hgh n subset 4. From the ab., CSRC s better than CRC for testng the llumnated mages, whch means the deleton mechansm makes the classfcaton more dscrmnatve. In addton, compared wth PSR, CSRC has about % hgher recognton result than t. he reason for whch s that the tranng samples from the correct class as probe s easy removed n the two-phases deleton, as well as the samples wth negatve coeffcent are not deleted. o the contrary CSRC reduces the rsk that the correct tranng samples wll be removed n one tme through the mult-phase compettve deleton. ORL database ORL database, created by A& lab n Cambrdge Unversty, contans 400 face mages of 40 subjects,.e. each ndvdual provdng 0 face mages, ncludng expresson varants, multple drectons of posture change wthn 0% of the scale of the change. Dmensonalty of each face s reduced to 50 40. All face mages are show n Fg. 4. For Fg.. Frontal faces wth emoton and llumnaton changes on AR database. he top seven faces are from sesson one and the down seven samples are from sesson two ABLE I. RECOGNIION RAES (%) ON AR DSABASE Methods CRC PSR SRC CSRC Recognton rate 9.57 9.857 8.74 9 Fg. 3. Some sample faces of a subject from Extended Yale B database. he top row: seven mages wth moderate llumnance varatons from subset. he down row: a part mages wth large llumnaton varatons from subset 4 5 P a g e
Vol. 6, No. 8, 05 ABLE II. RECOGNIION RAES (%) ON EXENDED YALE B DAABASE methods CRC PSR SRC CSRC Subset 4 74.94 55.76 44.4 77.49 each subject, we choose the frst fve mages as the tranng mages and the rest mages are used for testng. From the ab. 3, the recognton results of these four methods are close. Snce the deleton operaton n CSRC, the sparse codng vector has more dscrmnaton than CRC, so CSRC has.5% hgher recognton rate than CRC. ABLE IV. RECOGNIION RAES (%) ON AR DAABASE FOR SUNGLASSES AND SCARF Methods PSR CSRC Sunglasses 56 6 Scarf 5.5 78.5 Fg. 4. en mages of a specfc class from the ORL database. he top row represents the tranng samples and the mages n the bottom row are testng samples ABLE III. RECOGNIION RAES (%) ON ORL DAABASE methods CRC PSR SRC CSRC Recognton rate 87 9.5 88 88.5 B. Recognton wth sunglasses and scarf In ths secton we test CSRC s ablty to cope wth real possbly malcous occlusons usng a subset of AR database. he chosen subset conssts of 00 mages of 00 subjects, 50 male and 50 female. For each subject, eght frontal faces (half face are from sesson one and another half are from sesson two) wthout occlusons are used as tranng samples. We select the testng face mages wth sunglasses (two samples for each subject and each sample wth about 0 percent occluson) and scarf (two samples for each subject and each sample wth approxmately 40 percent occluson on the faces) respectvely and the testng samples of a specfc subject are show n Fg. 5. he recognton rates by PSR and CSRC are shown n ab. 4. CSRC has lttle better than PSR for testng samples wth sunglasses. In the scarf case, the recognton rate by CSRC s 6% hgher than PSR s. Because that the proporton of the scarf almost reaches to 40%, t s lkely that the mages of correct class as probe are deleted n the frst collaboratve representaton n PSR. On the contrary, CSRC makes sure the mages of correct class of probe have hgh compettve. Fg. 5. Face mages wth sunglasses and scarf respectvely on AR database V. CONCLUSIONS In ths paper, a compettve representaton framework s proposed to solve the sparse representaton problem. he lowest compettve deleton mechansm ensures the compettve ablty for representng a probe decrease and enhances the compettve ablty of the correct class as the probe. What s more the fast algorthm makes the FR system more effcency. Accordng to the experments, the multple rounds of compettve representaton has better performance n general than the two-phase deleton. In addton, SCRC adoptvely reduces the over-fttng ssue of the regresson model. However, CSRC also has some dsadvantages, such as has not enough robustness to deal wth occlusons, dsguses, and corrupton. In the further, we wll pay more attenton on these dsadvantages. ACKNOWLEDGMEN hs work was supported partally by the Natonal Nature Scence Commttee of Chna under Grant Nos. 6076 and 6403053 and partally by Chongqng basc and fronter research project under Grant Nos. cstc04jcyja4008. REFERENCES [] J. Wrght et al., Robust Face Recognton va Sparse Representaton, Ieee ransactons on Pattern Analyss And Machne Intellgence, vol. 3, no., pp. 0-7, Feb, 009. [] M. Yang et al., Regularzed robust codng for face recognton, IEEE rans Image Process, vol., no. 5, pp. 753-66, May, 03. [3] M. Yang et al., Robust Sparse Codng for Face Recognton, 0 Ieee Conference on Computer Vson And Pattern Recognton (Cvpr), pp. 65-63, 0, 0. [4] L. Zhang et al., Collaboratve representaton based classfcaton for face recognton, arxv preprnt arxv:04.358, 0. [5] A. Y. Yang et al., "Fast l -mnmzaton algorthms and an applcaton n robust face recognton: A revew." pp. 849-85. [6] M. Yang, and L. Zhang, "Gabor feature based sparse representaton for face recognton wth gabor occluson dctonary," Computer Vson ECCV 00, pp. 448-46: Sprnger, 00. [7] N. Kwak, Prncpal component analyss based on l-norm maxmzaton, IEEE rans Pattern Anal Mach Intell, vol. 30, no. 9, pp. 67-80, Sep, 008. [8] J. Wang et al., Robust face recognton va adaptve sparse representaton, IEEE rans Cybern, vol. 44, no., pp. 368-78, Dec, 04. [9] Y. Xu et al., A wo-phase est Sample Sparse Representaton Method for Use Wth Face Recognton, Ieee ransactons on Crcuts And Systems for Vdeo echnology, vol., no. 9, pp. 55-6, Sep, 0. 6 P a g e
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