Real Time Robust L1 Tracker Using Accelerated Proximal Gradient Approach

Transcription

1 Rel Time Robust 1 Trcker Using Accelerted Proximl Grdient Approch Chenglong Bo 1,YiWu 2, Hibin ing 2, nd Hui Ji 1 1 Deprtment of Mthemtics, Ntionl University of Singpore, Singpore, Deprtment of Computer nd Informtion Sciences, Temple University, Phildelphi, PA, USA,19122 {bochenglong,mtjh}@nus.edu.sg, {wuyi,hbling}@temple.edu Abstrct Recently sprse representtion hs been pplied to visul trcker by modeling the trget ppernce using sprse pproximtion over templte set, which leds to the so-clled 1 trckers s it needs to solve n l 1 norm relted imiztion problem for mny times. While these 1 trckers showed impressive trcking ccurcies, they re very computtionlly demnding nd the speed bottleneck is the solver to l 1 norm imiztions. This pper ims t developing n 1 trcker tht not only runs in rel time but lso enjoys better robustness thn other 1 trckers. In our proposed 1 trcker, new l 1 norm relted imiztion model is proposed to improve the trcking ccurcy by dding n l 2 norm regulriztion on the coefficients ssocited with the trivil templtes. Moreover, bsed on the ccelerted proximl grdient pproch, very fst numericl solver is developed to solve the resulting l 1 norm relted imiztion problem with gurnteed qudrtic convergence. The gret running time efficiency nd trcking ccurcy of the proposed trcker is vlidted with comprehensive evlution involving eight chllenging sequences nd five lterntive stte-of-the-rt trckers. 1. Introduction Visul trcking hs been n ctive reserch topic in the computer vision community s it is widely pplied in the utomtic object identifiction, utomted surveillnce, vehicle nvigtion nd mny others. Despite gret progresses in lst two decdes, due to numerous fctors in rel life, mny chllenging problems still remin when designing prcticl visul trcking system. For exmple, sophisticted object shpe or complex motion, illution chnges nd occlusions ll my cuse serious stbility issues for visul trcker (see more detiled discussion in [26]). Recently, sprse representtion nd compressed sensing techniques (e.g. [5, 7]) for finding sprse solution of n under-detered liner system hve drwn gret del of ttention in both mthemtics nd mny pplied fields, including visul trcking [15, 16, 11, 14, 24]. Similr to sprsity-bsed pproch for fce recognition developed in [22], these trcking methods express trget by sprse liner combintion of the templtes in the templte spce, i.e., the trget is well pproximted by the liner combintion of only few templtes. Benefitting from the stble recovery cpbility of sprse signl using the l 1 norm imiztion (e.g. [5]), these trckers hve demonstrted good robustness in vrious trcking environments. In the 1 trcker first proposed by [15], hundreds of l 1 norm relted imiztion problems need to be solved for ech frme during the trcking process. The solver for the l 1 norm imiztions used in [15] is bsed on the interior point method which turns out to be too slow for trcking. A iml error bounding strtegy is introduced [16] to reduce the number of prticles, equl to the number of the l 1 norm imiztions for solving. A speed up by four to five times is reported in [16], but it is still fr wy from being rel time. An efficient solver for the l 1 norm relted problems hs been the key to use the 1 trcker in prctice. Moreover, in the existing 1 trcker, trivil templtes re included in the templte dictionry such tht its sprse liner combintion will present the occlusions nd imge noise in the trget. However, s we empiriclly observed, the sprse liner combintion of the trivil templtes sometimes include prts of the object in the trget, which will result in loss of trcking ccurcy in some sequences. Built upon the sme frmework of the 1 trcker [15, 16], this pper ims t developing more robust 1 trcker which runs in rel time. There re two min contributions in the proposed pproch. One is the introduction of newl 1 norm relted imiztion model which empiriclly showed improvements on the trcking ccurcy over the model used in [15]. The other more significnt contribution is the introduction of very fst numericl method to solve the resulting l 1 norm imiztion problems which leds to rel time 1 trcker. It is noted tht the l 1 imiztion problem shown in [15] is just specil cse of our l 1 imiztion problem. Thus, the proposed numericl method cn lso be pplied to the originl 1 trcker to

2 mke it rel-time trcker. 2. Relted Work Among mny pproches for rel world visul trcking problem, discritive trcking nd genertive trcking re two different ctegories with different formultions. Trcking problem is formulted s binry clssifiction problem in discritive trcking methods. Discritive trckers locte the object region by finding the best wy to seprte object from bckground; see e.g. [1, 2, 21, 27]. In [1], feture vector is constructed for every pixel in the reference imge nd n dptive ensemble of clssifiers is trined to seprte pixels tht belong to the object from the ones in the bckground. Online multiple instnce lerning is used in [2] to chieve robustness to occlusions nd other imge corruptions. Sprse Byesin lerning is used in [21]. Globl mode seeking is used in [27] to detect the object fter totl occlusion nd reinitilize the locl trcker. Genertive trcking method is bsed on the ppernce model of trget object. Trcking is done vi serching trget loction with best mtching score by some metric; see e.g. eigentrcker [3], men shift trcker [6], incrementl trcker [18] nd covrince trcker [17]. To dpt to pose nd illution chnges of the object, ppernce model is often dynmiclly updted during the trcking. Sprse representtion hve been pplied to trcking problem in [15], nd lter exploited in [14, 13]. In [15], trcking cndidte is sprsely represented by trget templtes nd trivil templtes. In [14], group sprsity is integrted nd very high dimensionl imge fetures re used for improving trcking robustness. In these pproches, the sprse representtion is obtined vi solving l 1 -norm relted imiztion problem [15] or l -norm relted imiztion in [14, 13]. It is well known tht l -norm relted imiztion is n NP-hrd problem. The lrge-scle l 1 -norm relted imiztion is lso chllenging problem due to the non-differentibility of l 1 norm. The numericl methods for solving l 1 -norm relted imiztion in [15] is bsed on the interior point method [1], which is very slow when solving lrge-scle l 1 -norm imiztions. In recent yers, there hve been gret progresses on fst numericl methods for solving lrge-scle l 1 -norm relted imiztion problems rising in imge science, such s inerized Bregmn itertion [4], Split Bregmn method [8] etc. Menwhile, Yng et l. [25] hs done comprehensive study of the l 1 norm relted imiztion on robust fce recognition. Among ll these methods, one promising pproch is the so-clled ccelerted proximl grdient (APG) method introduced by [2] for imizing the summtion of one smooth function nd one non-differentil function. The APG method is used in [19] to solve unconstrined l 1 norm relted problem relted to imge restortion. 3. Introduction to 1 Trcker Our trcker is closely relted to the 1 trcker proposed by Mei nd ing [15]. The min differences lie in different imiztion model nd much fster numericl solver for the resulting l 1 norm imiztion problems. We first give brief review on the 1 trcker within the prticle filter frmework proposed in [16, 15]. Prticle Filter: The prticle filter provides n estimte of posterior distribution of rndom vribles relted to Mrkov chin. In visul trcking, it gives n importnt tool for estimting the trget of next frme without knowing the concrete observtion probbility. It consists of two steps: prediction nd updte. Specilly, t the frme t, denote x t which describes the loction nd the shpe of the trget, y 1:t 1 = {y 1, y 2,, y t 1 } denotes the observtion of the trget from the first frme to the frme t 1. Prticle filter proceeds two steps with following two probbilities: p(x t y 1:t 1 )= p(x t x t 1 )p(x t 1 y 1:t 1 )dx t 1, (1) p(x t y 1:t )= p(y t x t)p(x t y 1:t 1 ). (2) p(y t y 1:t 1 ) The optiml stte for the frme t is obtined ccording to the mximl pproximte posterior probbility: x t = rg mx x p(x y 1:t ). The posterior probbility (2) is pproximted by using finite smples S t = {x 1 t, x 2 t,, x N t } with different weights W = {w 1 t, w 2 t,, w N t } where N is the number of smples. The smples re generted by sequentil importnce distribution Π(x t y 1:t, x 1:t 1 ) nd weights re updted by: w i t w i p(y t x i t)p(x i t x i t 1) t 1 Π(x t y 1:t, x 1:t 1 ). (3) In the cse of Π(x t y 1:t, x 1:t 1 ) = p(x t x t 1 ), the eqution (3) hs simple form w i t w i t 1p(y t x i t). Then, the weights of some prticles mybe keep incresing nd fll into the degenercy cse. To void such cse, in ech step, smples re re-smpled to generte new smple set with equl weights ccording to their weights distribution. Sprse Representtion: The sprse representtion model ims t clculting the observtion likelihood for smple stte x t, i.e. p(z t x t ). At the frme t, given the trget templte set T t =[t 1 t, t 2 t,, t n t ], let S t = {x 1 t, x 2 t,, x N t } denote the smpled sttes nd let O t = {y 1 t, y2 t,, yn t } denote the corresponding cndidte trget ptch in trget templte spce. The sprse representtion model is then: y i t = T t i T I i I, y i t O t, (4) where I is the trivil templte set (identity mtrix) nd i t = [ i T ; i I ] is sprse. Additionlly, nonnegtive constrints re

3 imposed on i T for the robustness of the 1 trcker [15]. Consequently, for ech cndidte trget ptch y i t, the sprse representtion of y i t cn be found vi solving the following l 1 -norm relted imiztion with nonnegtive constrints: 1 2 yi t A 2 2 λ 1,, (5) where A =[T t,i, I]. Finlly, the observtion likelihood of stte x i t is given s p(z t x i t)= 1 Γ exp{ α yi t T tc i T 2 2}, (6) where α is constnt controlling the shpe of the Gussin kernel, Γ is norml fctor nd c i T is the imizer of (5) restricted to T t. Then, the optiml stte x t of frme t is obtined by x t =rgmxp(z t x i t). (7) x i t St In ddition, templte updte scheme is dopted in [15] to overcome pose nd illution chnges. Miniml Bound: In [15], the l 1 -norm relted imiztion problem (5) is solved by the interior point method which is very slow. A iml error bounding method is then proposed in [16] to reduce the number of needed l 1 imiztions. Actully, their method is bsed on the following observtion: T t y 2 2 T t â y 2 2, R N, (8) where â =rg T t y 2 2. (9) Consequently, for ny smples x i t, its observtion likelihood hs the following upper bound: p(z t x i t) 1 Γ exp{ α T tâ y i t 2 2} q(z t x i t), (1) where q(y i t xi t) is the probbility upper bound for stte x i t.it is seen tht if q(z t x t ) < 1 i 1 2N p(z t x j t), then the smple j=1 x i t will not pper in the resmple set. In other words, x i t cn be discrded without being processed. Thus, twostge resmple method is proposed in [16] to significntly reduce the number of smples needed in trcking. 4. Rel Time 1 Trcker Even though the iml error bound [16] ws proposed to reduce the computtion lod for 1 trcker, there re still mny l 1 -norm relted imiztions for solving during the trcking process, For exmple, in the sequence cr with 62 frmes, round 8, l 1 -norm relted imiztions (5) needs to be solved with iml error bound resmpling scheme in [16]. Therefore, the speed bottleneck in the 1 trcker is how to solve the l 1 -norm relted imiztion (5) much fster, in the scle of hundreds of times. Also, s seen in the model (5), the trivil templtes re included in the templte dictionry such tht its sprse liner combintion will represent the occlusions nd imge noise in the trget. However, s we observed in the experiments, the sprse liner combintion will sometimes include prts of the object in the trget which my led to loss of trcking ccurcy in some sequences. In this section, we first proposed modified version of the imiztion problem (5) such tht the sprse liner combintion of trivil templtes cn represent the occlusions nd imge noise more ccurtely. Then, bsed on the ccelerted proximl grdient pproch [2], we proposed fst numericl method for solving the resulting l 1 norm relted imiztion problem such tht the trcker runs in rel time. It is noticed tht the developed method is lso pplicble to originl imiztion problem in (5) A modified l 1 norm relted imiztion model There re two types of templtes in the templte dictionry used by (5): trget templtes nd trivil templtes. The trget templtes re updted dynmiclly for representing trget objects during the trcking process. The trivil templtes (identity mtrix I) is for representing occlusions, bckground nd noise. However, since prts of objects my lso be represented by the trivil templtes, the region detected by the originl trcker sometimes does not fit the trget very ccurtely. We tke modified version of (5) for improving trcking ccurcy. The new model is bsed on the following observtion. When there re no occlusions, the trget in the next frme should be well pproximted by sprse liner combintion of trget templtes with smll residul. Thus, the energy of the coefficients in ssocite with trivil templtes, nmed trivil coefficients, should be smll. On the other hnd, when there exist noticeble occlusions, the trget in the next frme cnnot be well pproximtion by ny sprse liner combintion of trget templtes, the lrge residul (corresponding to occlusions, bckground nd noise in n idel sitution) will be compensted by the prt from the trivil templtes, which leds to lrge energy of the trivil coefficients. The imiztion (5) is obviously not optiml since it does not differentite these two cses. In other words, to optimize the usge of the trivil templtes in the trcking, we need to dptively control the energy of the trivil coefficients. Tht is, when occlusions re negligible, the energy ssocited with trivil templtes should be smll. When there re noticeble occlusions, the energy should be llowed to be lrge. This motivtion leds

4 where F () is differentible convex function with ipschitz continuous grdient 1 nd G() is non-smooth but convex function. The outline of the APG method is given in Algorithm 1. The efficiency of the APG method is justified by its qudrtic convergence; see Theorem 4.1. However, we emphsize here tht the APG method is fst only for prticulr type of function G. During ech itertion of Algorithm 1, we need to solve imiztion in Step 2. So, the qudrtic convergence of APG is mterilized only when the sub-problem in Step 2 hs n nlytic solution. Theorem 4.1 ([2]) et {α k } is the sequence generted by Algorithm 1. Then within K = O( /ɛ) itertions, {α k } chieves ɛ-optimlity such tht α K α <ɛ,where α is one imizer of (12) Figure 1. Illustrtion of the 1 trcker on the sequence lemg using the model (5) nd the 1 trcker using the proposed model (11). The first nd the second row: results using (5) nd using (11) respectively. st row: the energy rtio I 2/ 2. The left grph is from (5) nd the right is from (11). to the following imiztion model for 1 trcker 1 2 y A 2 2λ 1 μ t 2 I 2 2, s.t. T, (11) where A =[T t,i], =[ T ; I ] re the coefficients ssocited with trget templtes nd trivil templtes respectively, nd the prmeter μ t is prmeter to control the energy in trivil templtes. In our implementtion, the vlue of μ t for ech stte is utomticlly djusted using the occlusion detection method [16]. Tht is, if occlusions re detected, μ t =; otherwise μ t is set s some pre-defined constnt. The benefit of the dditionl l 2 norm regulriztion term I 2 2 is illustrted in Fig. 1. In Fig. 1, bout 3 percent of object energy is contined in trivil templtes from imiztion (5). In other words, trivil templtes cn not distinguish the object nd bckground. On the other hnd, we cn see the trivil templtes coefficients from imiztion (11) re smll nd led to better trcking results. At lst, we note tht the originl imiztion (5) is specil cse of the imiztion (11) by setting μ t = Fst numericl method for solving (11) The proposed method for solving the imiztion problem (11) is bsed on the ccelerted proximl grdient (APG) pproch [2]. APG pproch. The APG method is originlly designed for solving the following unconstrined imiztion: F ()G(), (12) Algorithm 1 the generic APG pproch in [2] (i) Set α = α 1 = R N nd set t = t 1 =1. (ii) For k =, 1,..., iterte until convergence β k1 := α k t k 1 1 t k (α k α k 1 ); α k1 := rg 2 β k1 F (β k1) 2 2 G(); t k1 := 1 14t 2 k 2. (13) Reformultion of (11) for pplying APG method. As we see, the originl APG method is designed for unconstrined imiztion problem which cn not be directly pplied to (11). Thus, we need to convert the constrined imiztion model into n unconstrined problem. et 1 R N denote the vector with ll entries re equl to 1 nd let 1 R N () denote the indictor function defined by {, ; 1 R N () = (14), otherwise. It is esy to see tht the imiztion (11) is equivlent to the following imiztion problem: rg 1 2 y A 2 2λ1 T T I 1 μ t 2 I 2 21 R n ( T ). (15) Then, the APG method cn be pplied to (15) with F () = 1 2 y A 2 2 λ1 T T μ t 2 I 2 2, G() = I 1 1 R n ( T ). (16) All steps in Algorithm 1 re trivil except Step 2, in which we need to solve n optimiztion problem: α k1 =rg 2 β k1 F (β k1) 2 2 G(). (17) 1 the grdient of F is ipschitz continuous if F (x) F (y) x y, x, y R N, for some constnt.

5 For generl function G, it cnnot be directly solved. However, in our setting, we hve the nlytic solution for (17); see Proposition 4.2. The lgorithm for solving l 1 -norm relted imiztion (11) is given in Algorithm 2. Proposition 4.2 If F () nd G() re defined in (16), then the imiztion problem (17) hs the following solution: α k1 T =mx(,g k1 T ) (18) α k1 I = T λ/ (g k1 I ). where g k1 = β k1 F (β k1) nd T is the softthresholding opertor: T λ (x) = sign(x) mx( x λ, ). Proof See the ppendix A. Tight ipschitz constnt estimtion. There is only one prmeter, the ipschitz constnt of F, is involved in Algorithm 2. This ipschitz constnt plys crucil role in the bove lgorithm. Algorithm 2 with n wrong will either diverges or converges very slowly. Next, we give tight upper bound of for F defined in (16) such tht is utomticlly set with optiml performnce; see Proposition 4.3. The detiled description of the proposed rel time 1 trcker, clled APG-1 trcker, is given in lgorithm 3. Proposition 4.3 et F denote the function defined in (16) with A =[T,I], where T is templte set nd I is the identity mtrix. The upper bound of the ipschitz constnt for F is given s follows. λ 2 mx μ t 1, (19) where λ mx is the lrgest singulr vlue of T. Proof See Appendix B. 5. Experiments Through the experiments, APG lgorithm is implemented with Mtlb, μ t = 5 in (11) when the occlusion is not detected nd otherwise, nd λ =1 2,T =8in Algorithm 2. Algorithm 3 APG-1 Trcker 1: Input: 2: Current frme F t ; 3: Smple Set S t 1 = {x i t 1} N i=1 ; 4: Templte set T = {t i } n i=1. 5: for i =1to N do 6: Drwing the new smple x i t from x i t 1; 7: Prepring the cndidte ptch y i t in templte spce; 8: Solving the lest squre problem (9); 9: Computing q i ccording to (1); 1: end for 11: Sorting the smples in descent order ccording to q; 12: Setting i =1nd τ =. 13: while i<nnd q i τ do 14: Solving the imiztion (11) vi Algorithm 2; Algorithm 2 Rel Time Numericl lgorithm for solving 15: Computing the observtion likelihood p i in (6); the imiztion (11) 16: τ = τ 1 2N p i; (i) Set α = α 1 = R N nd set t = t 1 =1. 17: i = i 1; 18: end while (ii) For k =, 1,..., iterte until convergence β k1 := α k t 19: Set p j =, j i. k 1 1 t k (α k α k 1 ); 2: Output: g k1 T := β k1 T (A (A β k1 y)) T / λ1 T ; 21: Finding the x g k1 I := β k1 I (A t ccording to (7); (Aβ k1 y)) I / 22: Detecting the occlusion [16] nd updte μ in (11); μβ k1 I /; 23: Updting the templte set T t 1 [16]; α k1 T := mx(,g k1 T ); 24: Updting the smple set S t 1 with p. α k1 I := T λ/ (g k1 I ); t k1 := (1 14t 2 k )/ Comprison with the 1 Trcker [16] The computtion efficiency nd trcking ccurcy of the proposed APG-1 trcker is first compred to tht of the BPR-1 trcker [16] on ten sequences. The verge running time of the proposed APG-bsed solver v.s. the interior point method used [16] is bout 1 : 15. As result, the verge running time of the APG-1 trcker v.s. the BPR- 1 trcker is round 1:2, with 6 prticles. The APG-1 trcker chieves bout verge 26 frmes per second with 6 prticles on PC with Intel i7-26 CPU (3.4GHz). The output bounding boxes of the trget from the two tckers re similr in mny sequences, while the results from APG-1 re more ccurte on some chllenging sequences. #277 #741 #676 #77 Figure 2. Demonstrtion of the improvement of APG-1 trcker (red) over BPR-1 (blue) on trcking ccurcy Qulittive Comprison with Other Methods The performnce of the proposed APG-1 trcker is lso evluted on eight publicly vilble video sequences nd is compred with five ltest stte-of-the-rt trckers

6 nmed Incrementl Visul Trcking (IVT) [18], Multiple Instnce erning (MI) [2], Visul Trcking Decomposition (VTD) [12], Incrementl Covrince Tensor erning (ICT) [23], nd Online AdBoost (OAB) [9]. The trcking results of the compred methods were obtined using the codes provided by the uthors with the defult prmeters nd using the sme initil positions in the first frme. The sequence jump ws cptured outdoors. The trget ws jumping nd the motion blurs re very severe. Results on severl frmes re presented in Fig. 3 (). The APG- 1 trcker, IVT, OAB, nd MI trcks the trget fithfully throughout the sequences. The other trckers fils trck the trget when there re brupt motion nd severe motion blur. The sequence cr shows vehicle undergoes drstic illution chnges s it psses beneth bridge nd under trees. Trcking results on severl frmes re shown in Fig. 3 (b). The APG-1 trcker nd IVT cn trck the trget well despite the drstic illution chnges, while the other trckers lose the trget fter it goes through the bridge. Results of the sequence singer re shown in Fig. 3 (c). In this sequence, we show the robustness of our lgorithm in severe illution chnges nd lrge scle vritions. Only our APG-1 trcker nd the VTD trcker cn trck the trget throughout the sequence. In the sequence womn (Fig. 3 (d)), only the APG-1 trcker is ble to trck the trget during the entire sequence. The other trckers drift to the mn when he occludes the trget due to his similr ppernce s the trget. In the sequence pole, person is wlking wy from the cmer nd is occluded by the pole for short time (Fig. 3 (e)). The IVT loses the trget from the strt nd the VTD strts to drift off the trget t frme 274 nd finlly loses the trget. All the rest successfully trck the trget but our APG-1 trcker recovers the trget scle better. Results on the sequence sylv re shown in Fig. 3 (f), where moving niml doll is undergoing chllenging pose vritions, lighting chnges nd scle vritions. The IVT, nd VTD eventully fils t frme 65 s result of drstic pose nd illution chnges. The rest trckers re ble to trck the trget for this long sequence while our APG-1 trcker performs with higher ccurcy. Results of the sequence deer re shown in Fig. 3 (g). In this sequence, we show the robustness of our lgorithm in bckground clutters nd the fst motion. Only our APG-1 trcker nd VTD cn trck the trget through the sequence. Fig. 3 (h) shows the results on the sequence fce. Mny trckers strt drifting from the trget when the mn s fce is severely occluded by the book. The APG-1 trcker nd IVT hndle this very well nd continue trcking the trget when the occlusion disppers. MI OAB ICT VTD IVT ours jump cr singer womn pole sylv deer fce Ave Tble 1. The verge trcking errors. The error is mesured using the Euclidin distnce of two center points, which hs been normlized by the size of the trget from the ground truth. The lst row is the verge error for ech trcker over ll the test sequences Quntittive Comprison with other methods To quntittively evlute the robustness of the APG-1 trcker under chllenging conditions, we mnully nnotted the trget s bounding box in ech frme for ll test sequences. The trcking error evlution is bsed on the reltive position errors (in pixels) between the center of the trcking result nd tht of the nnottion. As shown in Fig.4 nd Tble 1, the APG-1 trcker chieves comprble to the best performer on the sequence jump, singer nd fce to the best-performed trckers, nd on ll the other sequences it performs best. 6. Conclusion In summry, bsed on the frmework of the 1 trcker [15, 16], we developed rel time 1 visul trcker with improved trcking ccurcy. The ccurcy improvement is chieved vi new imiztion model for finding the sprse representtion of the trget nd the rel time performnce is chieved by new APG bsed numericl solver for the resulting l 1 norm imiztion problems. The experiments lso vlidted the high computtionl efficiency nd better trcking ccurcy of the proposed APG-1 trcker. References [1] S. Avidn. Ensemble trcking. In CVPR, 25. [2] B. Bbenko, M. Yng, nd S. Belongie. Visul trcking with online multiple instnce lerning. In CVPR, 29. [3] M. Blck nd A. Jepson. Eigentrcking: Robust mtching nd trcking of rticulted objects using view-bsed representtion. IJCV, 26(1):63 84, [4] J. Ci, S. Osher, nd Z. Shen. inerized bregmn itertions for compressed sensing. Mth. Comp, 78:55 59, 29. [5] E. Cndes, J. Romberg, nd T. To. Stble signl recovery from incomplete nd inccurte mesurements. Commu. on pure nd pplied mthemtics, 59(8): , 26. [6] D. Comniciu, V. Rmesh, nd P. Meer. Kernel-bsed object trcking. PAMI, 25(5): , 23.

7 #15 #16 #31 #135 #176 #311 () #174 #176 #293 #354 #418 #6 #8 #49 #85 #18 #133 #35 (b) (c) #177 #19 #197 #218 #373 #5 #14 #89 #236 #274 #372 #397 (d) (e) #23 #271 #358 #544 #65 #731 #8 #12 #27 #38 #51 #61 (f) (g) #81 #139 #251 #417 #643 #8 (h) Figure 3. Trcking results of different lgorithms for sequences jump(), cr(b), singer(c), womn(d), pole(e), sylv(f), deer(g) nd fce(h). [7] D. Donoho. Compressed sensing. IEEE Trns. on Informtion Theory, 52(4): , 26. [8] T. Goldstein nd S. Osher. The split bregmn method for l1 regulrized problems. SIAM J Img. Sci., 2: , 29. [9] H. Grbner, M. Grbner, nd H. Bischof. Rel-time trcking vi online boosting. In BMVC, 26. [1] S. Kim, K. Koh, M. ustig, S. Boyd, nd D. Gorinevsky. An interior-point method for lrge-scle l1-regulrized lest squres. IEEE J Sel Topics in Sig. Proc., 1:66 617, 27. [11] S. Kwk, W. Nm, B. Hn, J.H. Hn. erning Occlusion with ikelihoods for Visul Trcking. In CVPR, 211. [12] J. Kwon nd K. ee. Visul trcking decomposition. In CVPR, 21. [13] H. i, C. Shen, nd Q. Shi. Rel-time Visul Trcking Using Compressive Sensing. In CVPR, 211. [14] B. iu,. Yng, J. Hung, P. Meer,. Gong, nd C. Kulikowski. Robust nd fst collbortive trcking with two stge sprse optimiztion. In ECCV, 21.

8 # jump # cr # singer # womn # pole # sylv # deer # fce Figure 4. The trcking error for ech test sequence. The error is mesured the sme s in Tble 1 nd the legend s in Fig.3. [15] X. Mei nd H. ing. Robust visul trcking using l1 imiztion. In ICCV, 29. [16] X. Mei, H. ing, Y. Wu, E. Blsch, nd. Bi. Minimum error bounded efficient 1 trcker with occlusion detection. In CVPR, 211. [17] F. Porikli, O. Tuzel, nd P. Meer. Covrince trcking using model updte bsed on lie lgebr. In CVPR, 26. [18] D. Ross, J. im, R. in, nd M. Yng. Incrementl lerning for robust visul trcking. IJCV, 77(1): , 28. [19] Z. Shen, K. Toh, nd S. Yun. An ccelerted proximl grdient lgorithm for frme bsed imge restortions vi the blnced pproch. SIAM J on Img. Sci., 4:573, 211. [2] P. Tseng. On ccelerted proximl grdient methods for convex-concve optimiztion. SIAM J on Opti., 28. [21] O. Willims, A. Blke, nd R. Cipoll. Sprse byesin lerning for efficient visul trcking. PAMI, 27(8): , 25. [22] J. Wright, A. Yng, A. Gnesh, S. Sstry, nd Y. M. Robust fce recognition vi sprse representtion. PAMI, 31(1):21-227, 29. [23] Y. Wu, J. Cheng, J. Wng, H. u, J. Wng, H. ing, E. Blsch, nd. Bi. Rel-time Probbilistic Covrince Trcking with Efficient Model Updte. IEEE T-IP, 212. [24] Y. Wu, H. ing, J. Yu, F. i, X. Mei, nd E. Cheng. Blurred Trget Trcking by Blur-driven Trcker. In ICCV, 211. [25] A. Yng, A. Gnesh, S. Sstry, nd Y. M. Fst l1- imiztion lgorithms nd n ppliction in robust fce recognition: review. In ICIP, 21. [26] A. Yilmz, O. Jved, nd M. Shh. Object trcking: A survey. Acm Computing Surveys, 38(4):13, 26. [27] Z. Yin nd R. Collins. Object trcking nd detection fter occlusion vi numericl hybrid locl nd globl mode-seeking. In CVPR, 28. Appendix A: Proof of Proposition 4.2. The optimiztion problem (17) is expressed s follows, 2 g k R n ( T ) I 1. (2) Since the vribles of re independent, (2) is the sme s T 2 T g k1 T R n ( T ), (21) I 2 I g k1 I 2 2 λ I 1. It is esy to see the solution of first imiztion in (21) is the projection of g k1 T to the R n spce, i.e. mx(,g k1 T ). For the second imiztion in (21), ll the vribles re independent. So, we only need to solve the following imiztion : x 2 y x 2 2 λ x 1 f(x), (22) where x, y R. The imizer of (22) cn be expressed s soft thresholding opertion: x = T λ/ (y) =sgn(y) mx( y, ). (23) Thus, we hve I = T λ/ (g k1 I ) s the imizer of (21). Appendix B: Proof of Proposition 4.3. From (16), we hve ( ) 2 T F (x) = T T (24) T (1 μ)i Assume T = UΣV by singulr vlue decomposition, where U nd V re orthonorml mtrices, Σ R m N (m< N) with Σ ii = λ i nd λ 1 λ 2... λ m (. It is esy to know ) 2 F (x) is similr to M Σ Σ Σ. So λ Σ (1μ)I F mx = λ Mmx λ 2 mx 1μ, where λ F mx, λ Mmx nd λ mx re the lrgest singulr vlues of 2 F (x), M nd T respectively.