Collaboratively Regularized Nearest Points for Set Based Recognition

Transcription

1 WU, MINOH, MUKUNOKI: COLLABORATIVELY REGULARIZED NEAREST POINTS 1 Collaboratvely Regularzed Nearest Ponts for Set Based Recognton Yang Wu mm.meda.kyoto-u.ac.jp/members/yangwu Mchhko Mnoh mm.meda.kyoto-u.ac.jp/members/mnoh Masayuk Mukunok mm.meda.kyoto-u.ac.jp/members/mukunok Academc Center for Computng and Meda Studes Kyoto Unversty Kyoto, , Japan Abstract Set based recognton has been attractng more and more attenton n recent years, benefttng from two facts: the dffculty of collectng sets of mages for recognton fades quckly, and set based recognton models generally outperform the ones for sngle nstance based recognton. In ths paper, we propose a novel model called collaboratvely regularzed nearest ponts (CRNP) for solvng ths problem. The proposal nherts the merts of smplcty, robustness, and hgh-effcency from the very recently ntroduced regularzed nearest ponts (RNP) method on fndng the set-to-set dstance usng the l 2 -norm regularzed affne hulls. Meanwhle, CRNP makes use of the powerful dscrmnatve ablty nduced by collaboratve representaton, followng the same dea as that n sparse recognton for classfcaton (SRC) for mage-based recognton and collaboratve sparse approxmaton (CSA) for set-based recognton. However, CRNP uses l 2 -norm nstead of the expensve l 1 -norm for coeffcents regularzaton, whch makes t much more effcent. Extensve experments on fve benchmark datasets for face recognton and person re-dentfcaton demonstrate that CRNP s not only more effectve but also sgnfcantly faster than other state-of-the-art methods, ncludng RNP and CSA. 1 Introducton As takng pctures/vdeos and sharng them over networks get easer and more popularzed, collectng a set of mages for recognzng an object category/nstance becomes ncreasngly convenent. Therefore, the drecton of usng a set of nstances/mages together for recognton, namely set based recognton [10], has got rapdly growng attenton n recent years. Though most of the methods proposed for solvng ths problem have been only tested on face recognton [2, 10, 12, 13] and person re-dentfcaton [9, 10], they are generally applcable to any recognton tasks. Set based recognton models have the potental to outperform sngle nstance based recognton approaches under the same condtons, due to that more nstances for testng generally means a hgher probablty to extract dscrmnatve nformaton for correct classfcaton. However, the performance largely depends on the detaled desgn of the model. c The copyrght of ths document resdes wth ts authors. It may be dstrbuted unchanged freely n prnt or electronc forms.

2 2 WU, MINOH, MUKUNOKI: COLLABORATIVELY REGULARIZED NEAREST POINTS In the past few years varous approaches have been proposed, whch were revewed n [10] and [12]. In ths paper, we are focusng on the type of non-parametrc set-to-set dstance fndng approaches, whch have shown superor performance when compared wth other approaches. A smple model usng the mnmum pont-wse dstance (referred to as MPD for short) as the set-to-set dstance has been appled for person re-dentfcaton n [4] and got mpressve results. However, MPD s senstve to noses and outlers as the set-to-set dstance only depends on a sngle pont from each set. To overcome ths ssue, AHISD/CHISD (Affne/Convex Hull based Image Set Dstance) [2] computes an affne/convex hull for each set and then treats the geometrc dstance (dstance of closest approach) between two hulls as the set-to-set dstance. Snce such a dstance can be vewed as the dstance between two vrtual ponts generated from each of the two sets by lnear combnatons, t has some robustness to noses and outlers. Later on, SANP (Sparse Approxmated Nearest Ponts) [13] extended the model of CHISD by enforcng the sparsty of samples used for lnear combnaton, and showed better performance on face recognton than CHISD. Smlar to CHISD, there are also kernel versons of SANP (KSANP) as presented n [6], whch performed slghtly better than SANP. The work of SBDR (Set Based Dscrmnatve Rankng for Recognton) [10] opens a door towards ntegratng the power of unsupervsed set-to-set dstance fndng models wth the dscrmnatve ablty of supervsed learnng based approaches. Due to the dffculty of optmzng these two objectves n one step, SBDR adopts an teratve strategy to alternate setto-set dstance fndng (ether CHISD or SANP) and dscrmnatve metrc learnng. Though both SANP and SBDR (wth SANP) have shown qute mpressve results, they have a serous drawback of beng tme-consumng due to the l 1 -norm based sparsty term n ther model. Worse than SANP, SBDR needs an addtonal tranng stage whch takes much tme, as shown n the experments of ths paper. Very recently, the work of regularzed nearest ponts (RNP) [12] has ponted out that there s no need to use the complex formulaton wth many parameters and varables and the sparse constrant on the representaton coeffcents as adopted by SANP. Instead, RNP just uses the affne hull formulaton (as n AHISD [2]) regularzed by the l 2 -norm term, resultng n fewer parameters and varables. More attractvely, usng the l 2 -norm makes RNP have a closed-form soluton, thus beng much more effcent than SANP. Though beng smple, RNP performs even better than SANP on face recognton tasks. Despte ther dfferences n modelng, all of these approaches share the same dea of usng the ndependent set-to-set dstances between an arbtrary test/query set Q and each of the tranng/gallery sets X, {1,...,n} drectly for classfcaton, namely, dong nearestneghbor classfcaton based on these set-level dstances. The model of CSA (Collaboratve Sparse Approxmaton) [9], however, explores a new classfcaton model nspred by the success of sparse representaton for classfcaton (SRC) [8]. More concretely, CSA fnds the set-to-set dstance between Q and all X s together (treatng them as a large sngle set), and then t classfes Q by the reconstructon resduals usng only ndvdual gallery sets (.e. X s) and ther correspondng coeffcents. The collaboratve manner n the dstance fndng concdes wth the underlyng power of SRC as they both use all the tranng data to reconstruct a test set/sample. It tends to assgn hgher weghts to the samples belongng to the same class as the test set/sample, because statstcally those samples should be closer to the test set/sample than any others n a reasonably good feature space. Therefore, by dentfyng the set of samples whch have hgher coeffcents and a lower reconstructon resdual, we are lkely able to get the correct class label for the test set/sample. The dfferences between the ndependent dstance fndng approaches and the collaboratve dstance fndng approach

3 WU, MINOH, MUKUNOKI: COLLABORATIVELY REGULARIZED NEAREST POINTS 3 X 1 X 1 Q Q X Q Q X X n X n (a) Set-to-set dstances (b) Set-to-sets dstance Fgure 1: Independent dstance fndng v.s. collaboratve dstance fndng, usng the problem of person re-dentfcaton as an example (other tasks shall look the same). (a) The set-to-set dstances generated by tradtonal ndependent dstance fndng approaches; (b) the set-tosets dstance got by collaboratve dstance fndng methods. CSA are llustrated n Fgure 1. For set-to-sets dstance fndng, CSA utlzes the SANP model as t has the same l 1 -norm based constrant on the coeffcents as that n the SRC model. Snce t only needs to compute the set-to-sets dstance once, t s usually more effcent than the SANP approach tself whch has to compute n ndvdual set-to-set dstances. In ths paper, we propose a novel collaboratve dstance fndng approach called Collaboratvely Regularzed Nearest Ponts (CRNP). Unlke CSA, t uses RNP for set-to-sets dstance fndng, whch no longer ensures the sparsty of the coeffcents as SANP does, thus beng much smpler and computatonal more effcent than CSA. Compared wth RNP, CRNP needs only one-round set-level dstance fndng so t could be much faster than RNP. Moreover, CRNP enables ntroducng the dscrmnatve class-specfc (or set-specfc) coeffcents generated by collaboratve reconstructon nto ts classfcaton model, resultng n a further boostng of the performance. The effectveness of usng l 2 -norm nstead of l 1 - norm for classfcaton based on collaboratve reconstructon has also been wtnessed n the work of CRC (Collaboratve Representaton for Classfcaton) [14]. However, the proposed CRNP s to the best of our knowledge the frst one that deals wth set based recognton, and as a set-based model usng regularzed affne hulls, t s qute dfferent from smply extendng CRC to smultaneously handle a set of testng samples, whch wll be demonstrated n our experments. The rest parts of the paper are organzed as follows. Secton 2 brefly revews the RNP model, whch leads to our proposed model of CRNP to be detaled n secton 3. Secton 4 shows the expermental results on both face recognton and person re-dentfcaton usng 5 benchmark datasets. Conclusons and future work are gven n secton 5. 2 Regularzed Nearest Ponts RNP models each mage set by a regularzed affne hull (RAH) formulated as: RAH = { x = X α k α k = 1, α 2 σ }, (1)

4 4 WU, MINOH, MUKUNOKI: COLLABORATIVELY REGULARIZED NEAREST POINTS where the matrx X s a collecton of the samples belong to set wth each column denotng the feature vector of an mage and the vector α denotes the combnaton coeffcents. Note that n [12], the regularzer of RAH s defned usng a l p -norm to make the model as general as possble, but only p = 2 has been mplemented for RNP. Therefore, here we drectly use l 2 and keep t consstent throughout the paper. The dfference between RAH and AHISD s the addtonal l 2 -norm regularzaton of α, whch makes the affne combnaton only focus on the samples close to the sample mean of the set. For a gven test/query set Q and a tranng/gallery set X, RNP fnds two nearest ponts from the RAH of Q and the RAH of X, respectvely, by solvng the followng optmzaton problem: mn α,β Qα X β 2 2, s.t. k α k = 1, j β j = 1, α 2 σ 1, β 2 σ 2, (2) whose dual problem s mn α,β { Qα X β λ 1 α λ 2 β 2 2 }, s.t. k α k = 1, j β j = 1, (3) where λ 1 and λ 2 are Lagrangan multplers, and the affne hull constrants (.e., k α k = 1, j β j = 1) help avodng the trval soluton (α = β = 0). After gettng the soluton (α, β ) of Equaton 3, the set-to-set dstance between Q and X s defned to be d RNP = ( Q + X ) Qα X β 2 2, (4) where Q s the nuclear norm of Q,.e. the sum of the sngular values of Q. The nuclear norm term reflects the representaton ablty of a set, and t can remove the possble dsturbance unrelated to the class nformaton, to avod basng on large sets. Fnally, Q s classfed by C (Q) = argmn { d RNP }. (5) 3 Collaboratvely Regularzed Nearest Ponts The proposed CRNP model performs collaboratve dstance fndng usng all the tranng/gallery sets nstead of the ndependent set-by-set dstance fndng n RNP, so the dstance fndng model and ts optmzaton change accordng. Besdes that, CRNP has a dfferent classfcaton model whch makes use of the dscrmnatve coeffcents generated by collaboratve dstance fndng. The detals of these three aspects are gven as follows. 3.1 Collaboratve dstance fndng Gven the test/query set Q and all the tranng/gallery sets X, {1,...,n}, CRNP solves the followng optmzaton problem: { } Qα Xβ λ 1 α λ 2 β 2 2, s.t. k α k = 1, n =1 j β j = 1, (6) mn α,β where X = [X 1,...,X n ] denotes all the tranng/gallery sets together; β = [β T 1,..., β T n ]T are the correspondng coeffcents for these sets; λ 1 and λ 2 are trade-off parameters.

5 WU, MINOH, MUKUNOKI: COLLABORATIVELY REGULARIZED NEAREST POINTS 5 Ths problem nherts the dstance fndng model from RNP, however, the small change of replacng X n Equaton 3 by X causes a bg change from ndependent dstance fndng to collaboratve dstance fndng, whch may lead to qute dfferent classfcaton results, as llustrated n Fgure 1 and demonstrated n Secton Dstance fndng optmzaton The optmzaton problem 6 wth equalty constrants can be transformed to the followng unconstraned optmzaton problem: { mn Qα Xβ λ 1 α λ 2 β γ 1 α,β ( ) ) } 2 1 k α k + γ2 (1 n =1 j β j 2, where γ 1 and γ 2 are Lagrangan multplers. It can be rewrtten nto a smpler form as { mn z ˆQα ˆXβ } 2 α,β 2 + λ 1 α λ 2 β 2 2, (8) where z = [0 1,m, γ 1, γ 2 ] T wth m denotng the dmensonalty of the mage feature space, ˆQ = [Q T, γ 1 1 Nq,1,0 Nq,1] T wth N q denotng the number of samples n Q, and ˆX = [ X T, 0 Nx,1, γ 2 1 Nx,1] T wth N x denotng the number of samples n X. 0, j and 1, j denote the j zero matrx and the j dmensonal matrx of ones, respectvely. Though the above problem has a closed-form soluton, we follow [12] on alternatvely optmzng α and β, whch avods the tme-consumng matrx nverse operaton of an ntegrated matrx contanng both Q and X for each test/query set Q. In the alternatve optmzaton of problem 8, however, the matrx nverse operaton on the tranng data X s ndependent of Q, so t can be pre-computed before testng. More concretely, when α s fxed, β has a closed-form soluton (7) β = P x ( z ˆQα ), (9) where P x = ( ˆX T ˆX + λ 2 I ) 1 ˆX T (10) (wth I denotng the dentty matrx) only depends on X, so t can be pre-computed. When β s fxed, α also has a closed-form soluton α = P q ( z ˆXβ ), (11) where P q = ( ˆQ T ˆQ + λ 1 I ) 1 ˆQ T, wth I denotng the dentty matrx. The algorthm of CRNP for computng the nearest ponts from both tranng and test sets s summarzed n Algorthm 1. As clamed n [12], the objectve functon n Formula 8 has a lower bound of 0 and t s jontly convex w.r.t. α and β. Snce n the alternatve optmzaton, each step on updatng α and β decreases the objectve, the teraton wll converge to the global optmal soluton. In our experments to be presented, the teraton usually termnates n no more than 10 steps.

6 6 WU, MINOH, MUKUNOKI: COLLABORATIVELY REGULARIZED NEAREST POINTS Algorthm 1 COLLABORATIVELY REGULARIZED NEAREST POINTS (CRNP): Requre: The tranng/gallery sets X R m N x, an arbtrary test/query set Q R m N q, the pre-computed z, ˆX and P x (usng Equaton 10), and four trade-off parameters {λ 1,λ 2,γ 1,γ 2 }. Ensure: The representaton coeffcents for dstance fndng: α and β. 1: Construct ˆQ = [Q T, γ 1 1 Nq,1,0 Nq,1] T. 2: Compute the project matrx P q = ( ˆQ T ˆQ + λ 1 I) 1 ˆQ T. 3: Intalze β 0 = 1/N x. 4: whle not converged or not exceedng the maxmum number of teratons do 5: Update the representaton coeffcents: 6: α t+1 = P q (z ˆXβ t ). 7: β t+1 = P x (z ˆQα t+1 ); 8: end whle 9: Return α and β. 3.3 Classfcaton The collaboratve dstance fndng n CRNP mplctly makes β = [β 1,..., β n ] dscrmnatve. Therefore, we defne the dssmlarty between Q and X, {1,...,n} as dcrnp = ( Q + X ) Qα X β 2 2 / β 2 2, (12) where Q s the nuclear norm of Q,.e. the sum of the sngular values of Q. Then, Q s classfed by { } C (Q) = argmn d CRNP. (13) Compared wth drnp, d CRNP benefts from the dscrmnatve power of β, whch tends to make the class-specfc reconstructon resdual Qα X β 2 2 smaller and β 2 2 larger for the ground-truth label than any other labels j {1,...,n}, j. The effectveness of CRNP s classfcaton model wll be demonstrated n the followng secton. 4 Experments and Results To make the comparson wth the state-of-the-art approaches for set based recognton far and suffcent, we perform experments on both face recognton and person re-dentfcaton, and report results of all the related methods when applcable. Detaled expermental settngs are gven n secton 4.1, followed by the result comparson and analyss shown n secton 4.2. Fnally, we report the computatonal cost for the compared methods n secton Expermental settngs Datasets. For face recognton, we follow most of the related approaches on usng the Honda/UCSD dataset [7] and the CMU MoBo dataset [5] wth the same settngs of usng the frst 50 or 100 frames for recognton as n [13], [10] and [12]. The szes of the mages and the feature representatons exactly follow [10] (namely, raw pxels for Honda/UCSD dataset and LBP features for CMU MoBo dataset). Followng the common settngs, for

7 WU, MINOH, MUKUNOKI: COLLABORATIVELY REGULARIZED NEAREST POINTS 7 Honda/UCSD dataset, the specfed 20 sequences of 20 subjects are used for tranng/gallery and the rest 39 sequences are left for testng/queryng; for CMU MoBo dataset, we perform a 10-tme cross-valdaton by randomly choosng one sequence from the four canddates for each of the 24 subjects and have the unselected sequences left for testng. The propertes of the these two datasets can be found n [10] and [12]. For person re-dentfcaton, we follow [9] on usng the LIDS-MA and LIDS-AA datasets [1] for evaluaton, along wth an addtonal dataset CAVIAR4REID [3]. These datasets have nterestng and complementary propertes, thus beng nformatve for comparson. LIDS- MA and LIDS-AA contan multple mages (92 n LIDS-MA, and 21 to 243 n LIDS- AA) for each human ndvdual captured by two non-overlappng cameras (camera 1 and camera 3 n ther orgnal settng) at an arport wth large vewpont changes. The LIDS- MA dataset has 40 persons wth manually cropped mages, whle the LIDS-AA dataset contans as many as 100 ndvduals collected by an automatc trackng algorthm. Therefore, compared wth LIDS-MA, LIDS-AA s not only larger, but also noser wth localzaton errors and unequal class szes. For both datasets, we perform 10-tme cross-valdaton by randomly choosng 10 mages for each gallery/query set. Unlke LIDS-MA and LIDS-AA, CAVIAR4REID conssts of several sequences flmed n a shoppng centre. Besdes vewpont changes, t has broad resoluton changes and severe pose varatons. We follow [3] on tranng wth 22 specfed subjects and testng on the other 50 subjects. Each set (ether for gallery or query) contans 5 randomly sampled mages, and 10-tme cross-valdaton s performed for result averagng. We use the same 400-dmensonal color and texture hstograms based features as mentoned n [11] for all the three datasets. Methods for comparson. We compare CRNP wth all the related state-of-the-arts methods as ntroduced n secton 1, ncludng MPD[4], SRC[8], CRC[14], CHISD[2], SANP[13], KSANP[6], SBDR[10], CSA[9] and RNP[12]. Note that SRC and CRC n ths paper stand for the extended versons from ther orgnal models. We smply replace the sngle test mage n these two models by a set of mages to do an overall reconstructon and classfcaton. For KSANP and SBDR, we only report the results that have been lsted for the same tasks n ther orgnal publcatons, whle for the others, we have ther codes ether from ther authors (such as SRC, CHISD, SANP, and CSA) or mplemented by ourselves (such as MPD, CRC and RNP) run on the same data splts usng the same features. Parameters. For CRNP, we have λ 1 = λ 2 = 4,γ 1 = γ 2 = 1 fxed for all the experments. For the other methods, we used the recommended parameters gven n ther orgnal papers. 4.2 Expermental results and analyss Honda/UCSD dataset. The results for all the concerned methods are lsted n Table 1 for both the 50 frames/set and 100 frames/set settngs. Besdes the referred results for KSANP and SBDR, we also refer the orgnally reported results for some other methods when they are sgnfcantly dfferent from ther results n our experments. It s clear that the proposed CRNP greatly outperforms all the other methods (over 2.5%). For both expermental settngs, SRC, KSANP and RNP are relatvely better than the others. CMU MoBo dataset. The results for the CMU MoBo dataset are shown n Table 2. Unfortunately, the referred results for SBDR were generated wth the tranng set fxed to be the frontal sequence, whch s unlke the completely random sequence samplng n ths paper and also the other papers. Therefore, they are not counted for competng the best performance. In can be seen that CRNP stll sgnfcantly outperforms all the others when 50 frames/set s used, whle t s only slghtly worse than RNP when 100 frames are avalable for each set.

8 8 WU, MINOH, MUKUNOKI: COLLABORATIVELY REGULARIZED NEAREST POINTS Table 1: Face recognton accuracy (%) comparson on the Honda/UCSD dataset. The results wth stars are drectly coped from ther orgnal papers for reference, whle those wthout stars are got from our experments. The best results are shown n bold. Method MPD[4] SRC[8] CRC[14] CHISD[2] SANP[13] KSANP[6] SBDR[10] CSA[9] RNP[12] CRNP 50 frames / / / frames / / / Table 2: Face recognton accuracy (%) comparson on the CMU MoBo dataset. The results wth stars are drectly coped from ther orgnal papers for reference, whle those wthout stars are got from our experments. All these results are averaged over 10 random trals, wth the best ones shown n bold. Note that the referred results for SBDR were generated wth a dfferent expermental settng, so they are not counted for performance rankng. Method MPD[4] SRC[8] CRC[14] CHISD[2] SANP[13] SBDR[10] CSA[9] RNP[12] CRNP 50 frames / frames / It ndcates that too large set sze may weaken the dscrmnatve power of the collaboratve dstance fndng as t s more lkely that some samples from dfferent classes may confuse the correct class n collaboratve representaton. Even though, such a rsk s stll very low, whch demonstrates the robustness of CRNP. Person re-dentfcaton datasets. For person re-dentfcaton, snce t s wdely treated as a rankng problem and people usually care about the recognton accuracy n the top few ranks, we report both the rank-1 accuracy and the rank top 10% accuracy for a consderate comparson. As Table 3 shows, CRNP has sgnfcant superorty n both measures comparng wth the others, especally on the most challengng CAVIAR4REID dataset. The results of CRNP on CAVIAR4REID also greatly outperform the state-of-the-art results n [3]. Table 3: Performance comparson for person re-dentfcaton wth all the related methods on three benchmark datasets. Both the rank-1 accuracy and the rank top 10% accuracy (shown n parenthess). The results are averaged over 10 random trals, wth the best ones marked n bold. Dataset MPD[4] SRC[8] CRC[14] CHISD[2] SANP[13] CSA[9] RNP[12] CRNP LIDS-MA 50.0(75.0) 57.3(74.8) 28.5(50.0) 52.5(72.8) 46.8(74.8) 59.0(71.3) 53.3(76.0) 59.0(78.3) LIDS-AA 23.8(60.4) 36.0(68.9) 24.7(54.1) 24.6(58.2) 19.2(57.3) 22.5(59.6) 25.5(59.9) 35.4(71.6) CAVIAR4REID 19.0(47.2) 25.4(50.8) 16.6(37.6) 25.4(51.2) 25.2(52.4) 24.6(48.8) 24.0(50.2) 26.8(63.6) 4.3 Computatonal cost In addton to accuracy, we also evaluate the effcency of CRNP, n comparson wth the others. All the methods compared are mplemented n Matlab. We mplemented MPD, CRC, RNP and CRNP by ourselves wthout specfc code optmzaton. All the experments were conducted on a 2.67 GHz dual-core machne wth 20GB memory (actually no more than 2GB were used). Snce some of the methods can have (parts of) ther models pre-computed before testng, we report the pre-computaton tme for them n Table 4. The results show that the tranng phase of SBDR s very tme-consumng, whle the pre-computaton tme for other three methods are gnorable. In greater detal, CRNP needs a bt more tme than RNP, but less tme than CSA.

9 WU, MINOH, MUKUNOKI: COLLABORATIVELY REGULARIZED NEAREST POINTS 9 Table 4: For those methods whch can have (parts of) ther models pre-computed usng the tranng data, the total pre-computaton tme (n seconds) s lsted for comparson. Dataset Honda/UCSD CMU MoBo 50 frames 100 frames 50 frames 100 frames LIDS-MA LIDS-AA CAVIAR4REID SBDR[10] N/A N/A N/A CSA[9] RNP[12] CRNP Table 5: Computatonal cost comparson wth all the related methods on all of the recognton tasks. The results are averaged over 10 random trals f applcable, and we report them n the mllseconds per sample manner to elmnate the nfluence of dataset sze varaton. The best results for the methods excludng CRC are shown n bold. Dataset MPD[4] SRC[8] CRC[14] CHISD[2] SANP[13] SBDR[10] CSA[9] RNP[12] CRNP Honda/UCSD (50) Honda/UCSD (100) CMU MoBo (50) CMU MoBo (100) LIDS-MA N/A LIDS-AA N/A CAVIAR4REID N/A The testng tme for all the methods s lsted n Table 5, showng that CRNP s generally more effcent than all the others except CRC as t just performs MPD n a low-dmensonal projected space. 5 Concluson and Future Work We have proposed a novel set based recognton model whch s generally more effectve and sgnfcantly faster than other related methods. Expermental results on fve dfferent benchmark datasets have demonstrated ts superorty. A possble future work wll be ntroducng dctonary learnng nto the model to further mprove ts performance and robustness. Acknowledgment Ths work was supported by R&D Program for Implementaton of Ant-Crme and Ant- Terrorsm Technologes for a Safe and Secure Socety, Funds for ntegrated promoton of socal system reform and research and development of the Mnstry of Educaton, Culture, Sports, Scence and Technology, the Japanese Government. References [1] Slawomr Bak, Etenne Corvee, Franços Bremond, and Monque Thonnat. Boosted human re-dentfcaton usng remannan manfolds. Image and Vson Computng, 30 (6-7): , ISSN do: /j.mavs [2] Hakan Cevkalp and Bll Trggs. Face recognton based on mage sets. In CVPR, pages , 2010.

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