Kernel Collaborative Representation Classification Based on Adaptive Dictionary Learning

Transcription

1 Internatonal Journal of Intellgent Informaton Systems 2018; 7(2): do: /j.js ISSN: (Prnt); ISSN: (Onlne) Kernel Collaboratve Representaton Classfcaton Based on Adaptve Dctonary Learnng Zheng-png Hu 1, 2, Y Lu 2, 3, Xuan Zhang 2, 3, Yang-hua Yn 2, 3, Ru-xue Zhang 2, 3, De-gang Sun 1 1 Department of Electronc Informaton and Engneerng, Shandong Huayu Unversty of echnology, Dezhou, Chna 2 Department of Informaton and Engneerng, Yanshan Unversty, Qnhuangdao, Chna 3 Hebe Key Laboratory of Informaton ransmsson and Sgnal Processng, Qnhuangdao, Chna Emal address: o cte ths artcle: Zheng-png Hu, Y Lu, Xuan Zhang, Yang-hua Yn, Ru-xue Zhang, De-gang Sun. Kernel Collaboratve Representaton Classfcaton Based on Adaptve Dctonary Learnng. Internatonal Journal of Intellgent Informaton Systems. Vol. 7, No. 2, 2018, pp do: /j.js Receved: May 24, 2018; Accepted: August 30, 2018; Publshed: September 26, 2018 Abstract: In recent years, wth the progress of technology, face recognton s used more and more wdely n varous felds. he classfcaton algorthm based on sparse representaton has made a great breakthrough n face recognton. However, face mages are often affected by dfferent poses, lghtng, and expresson changes, so test samples are often dffcult to represent wth lmted orgnal tranng samples. Due to the conventonal dctonary learnng methods lackng adaptablty, we propose a kernel collaboratve representaton classfcaton based on adaptve dctonary learnng. In ths paper, the coarse to fne sparse representaton s related to the adaptve dctonary learnng problem. Frst, the labeled atom dctonary s learned from each knd of tranng samples by sparse approxmaton. Based on ths assumpton, we use an effcent algorthm to generate an adaptve dctonary that s related wth the test sample. hen, based on the adaptve class dctonary, the kernel collaboratve representaton s used to realze the nter class competton classfcaton. he kernel functon s combned wth the coarse to fne sparse representaton to extract the non-lnear factors such as facal expresson change, posture, llumnaton, occluson and so on. he kernel collaboratve representaton s used to realze the nter class competton classfcaton. he man advantage of ths approach s to combne coarse to fne kernel collaboratve representaton wth dctonary learnng to generate adaptve dctonares that approxmate to the test mage. Expermental results demonstrate that the proposed appraoch outperforms some prevous state-of-the-art dctonary learnng methods and sparse codng methods n face recognton. Keywords: Pattern Recognton, Dctonary Learnng, Kernel Space, Collaboratve Representaton 1. Introducton Recently, sparse codng and dctonary learnng are one of the most successful applcatons of pattern recognton and computer vson, and dctonary learnng based pattern recognton has been wdely concerned. Due to some face mages are affected by dfferent postures, llumnaton and expresson changes [1, 2], t s usually dffcult for test samples to be represented by orgnal tranng samples wth lmted number. Based on ths problem, dctonary learnng can effectvely use a large number of data to model the atttude, llumnaton, and facal expresson nformaton of the correspondng changes. herefore, the test sample can be better represented by the atoms from the optmzed dctonary. Wrght et al. [3] have demonstrated that the test mage can be approxmated by a sparse lnear combnaton of the tranng mages. he sparse coeffcents correspondng to most of the tranng samples are zero [4]. Fnally, we determne whch class the test samples belong to accordng to the mnmum reconstructon error of dfferent classes. It tres to fnd the potental contrbuton of dfferent tranng samples to test mages. All the tranng sample mages are used as the shared dctonary D. When the changes n the face mage are small or the tranng samples are full, the shared dctonary can fully capture the man features of face mage, so the dctonary can represent and classfy the test sample. Sparse representaton based classfcaton algorthm has made great breakthroughs n face recognton [5-8]. However, due to varous noses n the real applcaton, tranng samples usually have large ntra

2 16 Zheng-png Hu et al.: Kernel Collaboratve Representaton Classfcaton Based on Adaptve Dctonary Learnng class changes. It s usually dffcult for test samples to be represented by orgnal tranng samples wth lmted number of changes. At present, the research of face recognton based on sparse representaton s manly focused on two ponts: (1) dfferent sparse constrants; (2) dctonary optmzaton wth dfferent constrants. For sparse constraned problems, the sparse representaton based on l0-norm convex programmng algorthm can get preferably recognton results [9]. However, mnmzng l0 -norm s a NP problem wth large amount of computaton. he tradtonal sparse representaton algorthm model uses l1 -norm as a convex optmzaton functon approxmatng to l0 -norm to solve ths knd of sparsty problem [10]. However, there are stll problems wth the complexty of the algorthm and the relatvely hgh tme cost. Zhang L [11] proposed a collaboratve representaton wth the l2 -norm. Consderng that the tranng dctonary s suffcent, the recognton performance s smlar to the l1-norm. and t has a closed analytc soluton compared to the sparse representaton of the L1 norm, thus t has a lower computaton Complexty. Research shows that n sparse representaton classfcaton performance, nter-class collaboraton has more mportant contrbuton than sparsty. he authors n [12] gave a probablstc explanaton for collaboratve representaton and proposed probablstc cooperatve sparse representaton. Zheng J [13] proposed teratve constrant group sparse and adaptve weght learnng face recognton. It obtans more structural nformaton and dscrmnant nformaton compared wth other methods based on lnear regresson. he methods above are all based on the same basc assumpton that the test samples can be represented by a lnear combnaton of tranng samples. However, the orgnal face mages are nterfered by the nonlnear factors such as facal expresson change, posture, llumnaton, occluson and so on. hus, ths lnear hypothess does not explot the nonlnear relatonshp between the tranng samples, whch may reduce face recognton robustness. o classfy faces more effectvely, Xu Y [14] used nonlnear functons to transform the orgnal space to the feature space where the test mage can be approxmated by a sparse lnear combnaton of the tranng mages. Based on ths assumpton, kernel method s combned wth sparse representaton [15, 16] and low rank representaton [17, 18] to solve classfcaton and approxmaton problems. Dong Wang et al. proposed the Kernel Collaboratve Representaton (KCR) [19] to select face functon and combne Hammng Kernel and LBP features for face recognton, and the expermental results show that the method has a good recognton effect especally when the tranng sample s not suffcent. Based on the dfferent dctonary optmzaton problems, Rubnsten R [20] descrbed how to use mathematcs and learnng models to obtan the evoluton of the dctonary, whch showed the mportance of the dctonary learnng to the sparse representaton model. Besdes, Xu Y [21] provded a comprehensve overvew of face recognton dctonary learnng methods. o solve the occluson problem and mprove the robustness of face recognton, Yang M [22] used Gabor features to compress the dctonary, and proposed a dctonary model wth the sparse constrant framework. Amng at the sngle sample problem, We C [23] proposed the auxlary dctonary learnng algorthm to expand the orgnal dctonary and thus mprove the classfcaton performance. Hu Y S [24] used D-KSVD dctonary learnng method to get a dscrmnatve structured dctonary. In [25], a doman adaptve dctonary learnng algorthm was proposed to expand the ntra class dversty of the orgnal tranng samples by collaboraton wth the source data to solve the problem of vsual mage classfcaton n dfferent source domans. Xu et al. [21] frst selected tran samples that are near to the test samples. hen the test mage s estmated by the lnear combnaton of the selected tranng samples, and the recognzed result s obtaned. he experments show that the smple sparse representaton face recognton algorthm (NS) [26] can also obtan better performance. he above algorthm ndcated that the qualty of dctonary affected the performance of mage classfcaton. Besdes the above, face mages are affected by dfferent postures, lghtng and expresson changes. It s dffcult for test samples to be represented by lmted orgnal tranng samples. For each test mage, the best representaton dctonary may be dfferent. Consderng the conventonal dctonary learnng methods suffer from the problems of lackng adaptablty, we propose to construct an adaptve dctonary assocated wth the test sample. he labeled atom dctonary s learned from each knd of tranng samples by sparse approxmaton. Based on ths scheme, we could obtan an effcent algorthm to generate an adaptve dctonary whch related wth the test sample. Secondly, the coarse to fne sparse representaton s related to the adaptve dctonary learnng problem. We fully extracted the non-lnear factors such as facal expresson changes, posture, lghtng, and occluson that exsted n the face mage. he kernel collaboratve representaton s used to realze the nter class competton classfcaton. 2. Kernel Collaboratve Representaton Classfcaton Based on Adaptve Dctonary Learnng 2.1. Sparse Representaton Based Classfcaton Suppose that there are L ndvduals and each sample can be represented as a column vector. Let m n D = [ A, A, A ] R be a set of L ndvduals, 1 2 L A 1 2 n m n = [a,a,...,a ] R s a set of th ndvdual, where m = 1,2,, L, a R s the jth sample of the th tranng j sample and m s the dmenson of the tranng sample. n s the number of th tranng sample. n s the total number of samples. he purpose of sparse representaton classfcaton s that when a tranng sample set s gven, t can correctly dentfy whch category the test sample belongs to. Snce we can t

3 Internatonal Journal of Intellgent Informaton Systems 2018; 7(2): judge the label of test sample y, the test sample y can be represented by a lnear combnaton of all tranng samples D as: m y = Dx R (1) x s sparse coeffcent vector, and sparse coeffcent s obtaned by solvng lnear equaton. Where x = [0,,0, a, a,0, 0] R n. he non-zero,1, n coeffcent n x correspond that those assocated wth the th ndvdual. In face recognton, Face mages usually satsfy m < n, so the lnear equaton y = Dx s underdetermned. he soluton of the equaton s not unque. Sparse vectors only have a small number of elements nonzero, so the solutons are sparse. In order to solve the problem, the problem of norm optmzaton l 0 s adopted: ( l ): x = argmn x s.t. Dx = y (2) 0 0 Among them, l 0 norm s the number of nonzero elements of a coeffcent vector. However, Eq (2) s a NP-hard problem. It s dffcult to accurately solve. he exstng optmzaton theory shows that f x suffcent and sparse, the norm l 0 optmzaton problem can be transformed nto a norm l 1 optmzaton problem: ( l ): xˆ = arg mn x s.t. Dx = y (3) 1 1 However, the face mages collected n realty are nevtably affected by nose. It s usually dffcult to accurately represent test samples. hus, the exstence of error s permtted and the lmt of error tolerance s defned as ε. hus, Eq. (3) can be revsed as: ( l ): xˆ = arg mn x s.t. Dx-y ε (4) Fnally, the test sample y s classfed nto the mnmum resdual: dentty( y) = argmn y Aδ ( x ) (5) ˆ1 2 he authors [4] proposed a classfcaton algorthm based on collaboratve representaton. It was proved that the role of collaboraton between classes n representng the query sample s more mportant than the sparse constrants. Compared wth the l1 -regularzed sparse representaton based classfcaton (SRC), the l2-regularzed CRC_RLS has very compettve FR accuracy but wth sgnfcantly lower complexty. Here ˆρ can be obtaned: 2 2 { 2 2} ˆ ρ = argmn y Dρ + λ ρ (6) ρ where λ s the regularzaton parameter. he role of the regularzaton term s twofold. On the one hand, t guarantees the stablty of ˆρ, and then guarantees the sparsty of the coeffcent soluton. he soluton of CR wth regularzed least square n Eq. (6) can be easly and analytcally derved as: Let ( ) 1 ( D D λ ) 1 ˆ ρ = + I D y (7) P = D D + λ I D, obvously, P s ndependent of query y. So t can be pre-calculated as a projecton matrx P. Once a query sample y comes, we project y onto P va Py. he proposed CRC wth regularzed least square (CRC_RLS) algorthm s summarzed as follows. (1) Normalze the columns of D to have unt l 2 -norm. (2) Code y over D by Where ( ) 1 P = D D + λi D (3) Compute the regularzed resduals ˆ ρ = Py, (8) r = y A ˆ ρ ˆ ρ (9) 2 2 (4) Output the dentty of y as Identty y { r } ( ) arg mn 2.2. Kernel Collaboratve Representaton = (10) Kernel collaboratve representaton (KCR) face recognton algorthm frst uses the kernel functon to transform the orgnal space to the feature space. hen KCR represents the test samples through a lnear combnaton of all tranng samples n the feature space, and classfes the test samples based on the representaton results. Let A1,, A n denote n tranng samples n the orgnal space. Let Y be a test sample n the orgnal space. he nonlnear mappng functon φ s used to transform samples from orgnal space to kernel space. Suppose that test samples ϕ ( Y) n feature space can be approxmately by a lnear represented of tranng samples φ ( A1 ),, φ ( An ) n feature space: n ϕ( Y) = β ϕ( A) (11) = 1 Suppose that any sample n feature space s a column vector, we can rewrte Eq. (11) nto the followng equaton: ϕ ( Y) = ΦΨ (12) Where Φ = [ ϕ( A1) ϕ( A n )], Ψ = ( β1 β n ), snce Φ may not be a square matrx and ϕ s unknown, we cannot drectly solve Eq. (12). But, Eq. (12) can be expressed as follows: Φ ϕ( Y) = Φ ΦΨ (13)

4 18 Zheng-png Hu et al.: Kernel Collaboratve Representaton Classfcaton Based on Adaptve Dctonary Learnng j j he kernel functon s defned as k( A, A ) = ϕ ( A) ϕ( A ), Eq. (13) can be converted nto: Where K Y KY = KΨ (14) k( A1, Y) k( A1, A1) k( A1, An) =, K =, and k( A, Y) k( A, A) k( A, A ) n n 1 n n β1 Ψ = β n If K s a nonsngular matrx, Eq. (14) can be solved as follows: representaton and classfcaton based on adaptve dctonary learnng s proposed. Fg. 1 shows the flow chart of kernel collaboratve representaton and classfcaton based on adaptve dctonary learnng. he method manly ncludes two steps. Suppose there are L categores. Frst, the labeled atom dctonary s learned from each knd of tranng samples by coeffcent approxmaton. Based on ths scheme, we could obtan L fttng mages as adaptve dctonary whch related wth the test sample. Secondly, the test sample and the adaptve class dctonary D are projected to the kernel space, and the test samples are classfed by the lnear combnaton of the label adaptve dctonary n the kernel space. Each tranng sample and test sample was converted nto column vectors. 1 Ψ = K K Y (15) If K s sngular, Eq. (14) can be solved as: 1 µ Ψ = ( K + I) KY (16) Where µ s a postve constant and I s the dentty matrx. Eq. (16) can be represented as: K = ( K K ) Ψ = β K + + β K (17) Y 1 n 1 1 n n Where K = [ k( A1, A ) k( A2, A ) k( An, A )]. hs ndcates that the representng problem of test samples n the feature space has been converted nto a new problem of representng K y by K, K represents the th column of the matrx K. We refer to K as kernel vector of the th tranng sample. It seems that dfferent classes of tranng samples make dfferent contrbutons to representng K y. We evaluate the contrbuton of each class and classfy the test sample as follows: frst, we calculate the sum of the contrbuton of the tranng samples from each class. he tranng samples from the k th class are represented as A A. hus the test sample s lnearly represented by the S t k th tranng samples 2 g = β K + + β K he smaller the k s s t t error ek = KY gk s, the greater contrbuton of the kth class s. We dentfy the class that contrbutes the most to the test sample Y (that s, the class that corresponds to the mnmum error) and assgn Y to ths class. hs equvalent to classfy the test sample nto the class that s the most smlar to ths category, because the smallest resdual e k means that the lnear combnaton of the tranng samples of class k s closest to the test sample Kernel Collaboratve Representaton Classfcaton Based on Adaptve Dctonary Learnng Fgure 1. Flow chart for classfcaton algorthm of kernel sparse representaton based on adaptve dctonary learnng. Defne A = [ A1, A2, A L ] as a set of L ndvduals, m n A = [ a, a, a ] R s a set of th ndvdual,,1,2, n Where M s the sample dmenson and n s the number of tranng samples n each class. he steps of the kernel collaboratve representaton classfcaton based on adaptve dctonary learnng are as follows: (1) We code test sample y through each class of tranng samplesa, ˆ ρ = Py, where, P = ( A A + λi) A (2) We perform sparse approxmaton of test samples through each type of tranng sample d = A ˆ ρ, where, 1 = 1,2,, L. (3) Usng sparsely approxmated mages as adaptve m L dctonary D, D = [d 1,d 2,,d L] R. (4) Map the test sample y and the adaptve class dctonary D to the kernel space and solve the lnear system: K = KΨ Y Where, he orgnal sparse representaton uses all the tranng sample sets as a dctonary, the dctonary selecton s not flexble enough, and the tme complexty s hgh. Amng at the problem of dctonary optmzaton, a kernel collaboratve

5 Internatonal Journal of Intellgent Informaton Systems 2018; 7(2): k( d1, y) β1 K Y = Ψ = k( dl, y) β L k( d1, d 1) k( d1, dl) K = k( dl, d 1) k( dl, dl) If K s a nonsngular matrx, t can be solved as follows: 1 Ψ = K K Y If K s sngular, t can be solved by usng: 1 µ Ψ = ( K + I) KY (18) (5) he test sample s represented by the class k dctonary d. k where g k = β K (19) k k K = [ k(d,d ) k(d,d ) k(d,d )] (20) k 1 k 2 k L k (5) Classfyng test samples usng resduals: { } dentty( y ) = argmn k e k, e k = K Y g k (21) he kernel space classfcaton method has two advantages: (1) t proposes a kernel lnear system and uses t to classfy test samples; (2) the lnear system dentfcaton method has a lower tme complexty. Suppose m s the number of tranng samples for each class, there are L ndvduals, n = Lm s the total number of tranng samples. he kernel 2 representaton method should solve only one lnear system n 3 2 the form of Eq. (11), the tme complexty s O (n + n ). he l 1 sparse representaton uses an teratve method to solve the lnear system, whch has a hgher tme complexty. Even f we do not consder the teratve process and only solve the lnear system, the tme complexty of the l 1 sparse representaton 3 2 algorthm s O (n + n M+ nm), where M s the dmenson of the sample vector n the orgnal space. hus, the kernel space representaton method has a lower tme complexty than the l 1 sparse representaton method. 3. Expermental Results In order to verfy the effectveness and stablty of the proposed algorthm, We adopted the Gaussan kernel functon 2 k ( x, xj ) = exp x xj / ( 2 σ ), here σ s the parameter of the ernel functon. We perform the experments on the AR [27], FERE [28], and G [29] face databases. o verfy the effectveness of our approach, ADL NS [26] SRC were selected as benchmarks. Addtonally, all the comparson experments are conducted on a PC wth MALAB R2014a software Experment on AR Face Database he AR database contans 126 people I and each person has 26 mages. In ths paper, 50 women and 50 males were selected from the AR face database, each of whch contans 14 mages, ncludng changes n harstyle, expresson, and llumnaton. he sze of facal mage s and all the mages are down-sampled to pxels. Fg. 2 s an adaptve class dctonary obtaned by fttng a tranng sample on an AR face database wth 8 tranng samples per class. Fgure 2. AR database tranng samples ftted to adaptve class dctonary.

6 20 Zheng-png Hu et al.: Kernel Collaboratve Representaton Classfcaton Based on Adaptve Dctonary Learnng In ths subspace, N (=2, 4, 6, 8, 10) mages were selected as tranng samples n each class, and the rest mages were taken as test samples. able 1 shows the recognton rates of kernel collaboratve representaton classfcaton based on adaptve dctonary learnng (KADL), the sparse representaton classfcaton based on adaptve dctonary learnng (ADL), the smple fast sparse representaton (NS) and the sparse representaton classfcaton algorthm (SRC) wth the number of dfferent tranng samples, respectvely. From able 1, the recognton rate mproved wth the ncrease of tranng samples snce the number of tranng samples ncreases, dscrmnatng nformaton ncreases. In addton, our method fts each type of tranng sample and obtans more global nformaton than the NS method, and has a hgher recognton rate than SRC and ADL. able 1. Comparson of recognton rate of several methods on AR databases. Algorthms NS 75.25% 74.9% 76.88% 91.67% 91.75% SRC 78% 76.7% 80% 95.33% 97.25% ADL 77.08% 77.9% 80% 98% 99.75% KADL 77.17% 77.9% 82.25% 98.17% 99.75% We take the effect of mage dmensons nto account and performed the experments wth dfferent sample dmensons to demonstrate the robustness of the algorthm. o analyze the mpact of dmensonaltes on the methods, experments were performed wth dfferent sample dmensons to demonstrate the robustness of our algorthm. Fgure 3 shows the correct recognton rate change curve for dfferent recognton algorthms when the dmenson of mage features changes. It can be seen that the recognton rate of the proposed KADL approach s hgher than other algorthms as a whole. Recognton Rate (%) NS 55 SRC ADL KADL Feature Dmenson Fgure 3. Recognton rate curves of dfferent recognton methods under dfferent dmensons Experment on FERE face Database he experment was smulated on the FERE database. he FERE dataset contans a large number of face mages, and the mages from the same person's mages have dfferent ages, poses, expressons, and lghtng changes. hs experment uses a cut-out FERE face database, whch ncludes 200 ndvduals, each wth 7 face mages for a total of 1400 mages. he sze of all mages n the database s 80 80, All the mages are reszed to In each category, 5 mages were randomly selected as tranng samples, and the remanng 2 samples were used as test samples. Fgure 4 shows an adaptve class dctonary mage obtaned by fttng a tranng sample on a FERE face database. Fgure 4. FERE database tranng samples ftted to adaptve class dctonary.

7 Internatonal Journal of Intellgent Informaton Systems 2018; 7(2): able 2 shows the recognton rate of KADL, ADL, NS and SRC n the FERE database. able 2. Recognton accuracy on FERE face database for several algorthms. Algorthms NS SRC ADL KADL Recognton rate 62% 71% 70.75% 83.75% In order to further verfy the valdty of the KADL algorthm, experments are performed under dfferent feature dmensons. Fgure 5 shows the recognton rate curves of each algorthm when the feature dmensons of the mage are dfferent. From the data, we can see that the classfcaton effect of our proposed algorthm s better than other methods. Recognton Rate (%) NS SRC 40 ADL KADL Feature Dmenson Fgure 5. Recognton rate curves of dfferent recognton methods under dfferent dmensons Experment on G face Database hs secton uses the G face dataset to test the proposed approach. he G dataset contans 50 persons and each person has 15 mages. Some sample mages n G database are shown n Fgure 6. Fgure 6. Partal sample schematc n the G database. In the experment, N (= 7, 8, 9, 10) mages were randomly selected as tranng samples and the rest were used as test samples. All the mages are down-sampled to able 3 shows the recognton rates of KADL, ADL, NS and SRC wth dfferent number of tranng samples. able 3. Recognton accuracy on G face database for several algorthms. Algorthms NS 61.75% 66.86% 67.67% 68.4% SRC 68% 70.85% 73.66% 74.4% ADL 68.25% 71.14% 73% 76% KADL 69.5% 71.71% 73.33% 76.4% In order to further verfy the performance of KADL algorthm, we also performed experments on dfferent dmensons. Fgure 7 shows the recognton rate curves of dfferent methods under dfferent dmensons. From the Fgure 7, we can see that the results wth the proposed KADL approach are generally hgher than other methods. Recognton Rate (%) NS SRC 66 ADL KADL Feature Dmenson Fgure 7. Recognton rate curves of dfferent recognton methods under dfferent dmensons. 4. Conclusons In order to overcome the shortcomngs of tradtonal dctonary learnng mode, such as fxty and lackng of adaptablty, ths paper dscusses adaptve dctonary learnng. For each test mage, a proper dctonary model s constructed accordng to ts own characterstc, and an adaptve dctonary approxmates the nput pattern s generated. In order to fully extract the nonlnear factors such as facal expresson change, posture, llumnaton and occluson n face mages, test sample and adaptve dctonary are mapped to hgh-dmensonal feature space usng kernel functon, we classfy the test sample n the kernel space nstead of the orgnal space. he problem of face recognton s solved by combnng coarse and fne two step sparse representaton wth adaptve dctonary learnng. A seres of experments are carred out n the AR face database, the FERE face database and the G face database, whch prove the valdty of the kernel collaboratve representaton classfcaton based on adaptve dctonary learnng. Acknowledgements hs work s supported by Natonal Natural Scence Foundaton of Chna under Grant (No ), Natural

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