Efficient tracker based on sparse coding with Euclidean local structure-based constraint

WANG Hongyuan, ZHANG Ji, CHEN Fuhua

(1. School of Information Science and Engineering, Changzhou University, Changzhou, Jiangsu, China 213164; 2. Department of Natural Science and Mathematics, West Liberty University, West Virginia, United States 26074)

Efficient tracker based on sparse coding with Euclidean local structure-based constraint

WANG Hongyuan1, ZHANG Ji1, CHEN Fuhua2

(1. School of Information Science and Engineering, Changzhou University, Changzhou, Jiangsu, China 213164; 2. Department of Natural Science and Mathematics, West Liberty University, West Virginia, United States 26074)

Abstract：Sparse coding (SC) based visual tracking (l1-tracker) is gaining increasing attention, and many related algorithms are developed. In these algorithms, each candidate region is sparsely represented as a set of target templates. However, the structure connecting these candidate regions is usually ignored. Lu proposed an NLSSC-tracker with non-local self-similarity sparse coding to address this issue, which has a high computational cost. In this study, we propose an Euclidean local-structure constraint based sparse coding tracker with a smoothed Euclidean local structure. With this tracker, the optimization procedure is transformed to a small-scale l1-optimization problem, significantly reducing the computational cost. Extensive experimental results on visual tracking demonstrate the e�ectiveness and efficiency of the proposed algorithm.

Keywords：euclidean local-structure constraint; l1-tracker; sparse coding; target tracking

Citation：WANG Hongyuan, ZHANG Ji, CHEN Fuhua. Efficient tracker based on sparse coding with Euclidean local structure-based constraint[J]. CAAI Transactions on Intelligent Systems, 2016, 11(1): 136-147.

Recently, visual target tracking was widely used in security surveillance, navigation, human-computer interaction, and other applications[1-2]. In a video sequence, targets for tracking often change dynamically and uncertainly because of disturbance phenomena such as occlusion, noisy and varying illumination, and object appearance. Many tracking algorithms were proposed in the last twenty years that can be divided into two categories: generative tracking and discriminant tracking algorithms[1-2]. Generative algorithms (e.g., eigen tracker, mean-shift tracker, incremental tracker, covariance tracker[2]) adopt appearance models to express the target observations, whereas discriminant algorithms (e.g., TLD[3], ensemble tracking[4], and MILTrack[5]) view tracking as a classification problem, thus attempting to distinguish the target from the backgrounds. Here, we present a new generative algorithm.

Based on sparse coding (SC; also referred to as sparse sensing or compressive sensing)[6-7], Mei proposed an l1-tracker for generative tracking[8-9], addressing occlusion, corruption, and some other challenging issues. However, this tracker incurs a very high computational cost to achieve efficient tracking (see section 2.1 and Fig.1 for details), and the local structures of similar regions are ignored, which may cause the instability and even failure of the l1-tracker. Indeed, the sparse coefficients, for representing six similar regions (CR1-CR6) under ten template regions (T1-T10) with original l1-tracker, are diversified (Fig. 3). ConsideringCR1andCR4, for example, we can see that although the latter is almost the partial occlusion version of the former, their sparse representations are very different. TrackingCR4(the woman’s face) may fail, because the tracker is likely to incorrectly consider the regionT8(the book) as its target.

Contrary to expectations, Xu proved that a sparse algorithm cannot be stable and that similar signals may not exhibit similar sparse coefficients[10]. Thus, a trade-off occurs between sparsity and stability when designing a learning algorithm. In addition, instability in the l1-optimization problem affects the performance of the l1-tracker.

Lu developed a NLSSS-tracker (NLSSST) based on SC applying a non-local self-similarity constraint by introducing the geometrical information of the set of candidates as a smoothing term to alleviate the instability of the l1-tracker[11]. However, its low efficiency (even slower than the original l1-tracker, Table 4) restricts its applicability in real-time tracking. In this study, motivated by the robustness of the l1-tracker and stability of NLSSST, we propose a novel tracker, called ELSS-tracker (ELSST), that is both robust and efficient. The main contributions of this study are as follows:

1)An efficient tracker, i.e., ELSST, is developed by considering the local structure of the set of target candidates. In contrast to the Lu5s[11]and Mei5s-tracker[8-9], our tracker is more stable and sparse.

2)The proposed tracker shows excellent performance in tracking different video sequences with regard to scale, occlusion, pose variations, background clutter, and illumination changes.

The rest of this study is organized as follows: l1- and NLSSS-tracker are introduced in section 2; in section 3, we analyze the disadvantages of these two trackers and propose our tracker; experimental results with our tracker and four comparison algorithms are reported in section 4; the conclusion and future work are summarized in section 5.

1Related works

1.1Sparse coding and the l1-tracker

Sparse coding is an attractive signal reconstruction method proposed by Candes[6-7]that reconstructs a signal y∈Rm×1with an over-complete dictionary D∈Rm×(n+2m)withasparsecoefficientvectorc∈Rn×1.TheSCformulationcanbewrittenasthel0-norm-constrainedoptimizationproblemasfollows:

(1)

whichisNP-hard,where‖·‖Fdenotesthevector’sFrobeniusnorm(i.e.,l2-norm),and‖·‖0countsthenumberofnon-zeroelementsofthevector.Candesprovedthatthel1-norm‖·‖1isthetightestupperboundofthel0-norm‖·‖0,andthus,Eq.(1)canberewrittenasthefollowingl1-optimizationproblem[6-7]:

(2)

BasedonSC,Meipresentedanicel1-trackerforrobusttracking[8-9](Fig. 1).Consideringthatthetargetislocatedinthelatestframe,thel1-trackerisinitializedinthenewarrivalframeandNcandidateregionsaregeneratedwithBayesianinference(Fig. 1a,b).Withntemplateslearnedfromprevioustrackingand2mtrivialtemplates(mpositiveonesandmnegativeones,wheremisthedimensionof1Dstretchedimage,Fig. 1c),Eq.(2)canbesolved(Fig. 1d,e,f).Withpositiveandnegativetrivialtemplates,Meiaddedanon-negativeconstraintc≥0inEq.(2),withwhichthereconstructionerrorsofallcandidateregionswithSCcoefficientscanbeusedtodeterminetheweightsforeachcandidate,andtheobjectinthenewarrivalframecanbelocatedwiththesumoftheweightedcandidates.Thedictionariesupdatingstrategiescanbeseenin[8-9].

Fig.1　Original l1-tracker algorithm

1.2Non-local self-similarity based sparse coding for tracking (NLSSST)

Recently, Xu indicated the trade-off between sparsity and stability in sparse regularized algorithms[10]. Moreover, Yang pointed out the same A-optimization issue in pattern classification[12]. Based on the fact that lots of similar regions exist in allNcandidates generated by Bayesian inference, Lu proposed his tracker with the non-local self-similarity constraint as

(3)

(4)

Taking the solution of the l1-tracker from Eq.(2) as the initial coefficientsc0, Eq.(4) can be solved through iterative computations[11]. However, the high computational cost of the original l1-tracker and iterative procedure for maintaining the neighborhood constraints of sparse coefficients make NLSSST inefficient in achieving real-timing tracking. In contrast to Fig. 1, the schematic diagram of NLSSST presented in Fig. 2, includes an additional neighborhood constraint betweenyiandNK(yi).

Fig. 2　Lu’s NLSSST Algorithm

2Euclidean local structure-based sparse coding for tracking (ELSST)

To circumvent the heavy computation burden of the l1-tracker and NLSSST (Table 4), we propose an efficient tracker, called ELSST, that considers the local Euclidean structures of the candidates.

2.1Original euclidean local structure constraint sparse coding (Original ELSSC)

It is evident from Eq. (4) that NLSSST attempts to solve a double l1-norm problem. However, it is well known that the l2-norm is much more commonly used for measuring the distance between two vectors and is much easier to optimize than the l1-norm. Thus, we take the former to measure the relationships between the sparse coefficient vectors, which are close to each other, i.e., the Euclidean local-structure constraint, and the latter l1-norm ofCto maintain the sparsity of the optimization as follows:

(5)

Table 1　Optimization for ELS constraint based SC(ELSSC)

Equation (5) is the objective function of our Euclidean local structure constraint-based SC and can be solved through iterative computation. In particular, at thet-th iteration, for a single candidateyiinY, Eq. (5) can be written as follows:

(6)

(7)

whereλis convex. According to Daubechies[13], when λI-DTDisastrictlypositivedenitematrix,ψ(ci,c0)isstrictlyconvexforanyc0withrespecttoci.Hence,inourexperiments,theconstantλissetaccordingly(λ=γ- 2β;Table1).Oncetheover-completedictionaryDisfixed,wecanderivethefollowingconvexobjectivefunctionfromEq. (7):

(8)

where

and

(9)

To solve Eq. (9) using SVD, we decompose the over-complete dictionaryD∈Rm×(n+2m)asD=UΣVT,whereU∈Rm×m,Σ∈Rm×(n+2m)andV∈R(n+2m)×(n+2m). SinceVisanorthogonalmatrix,Eq. (9)canberewrittenas

(10)

2.2Improved euclidean local structure constraint sparse coding (Improved ELSSC)

IfminEq. (10)islarge,itistime-consumingtoobtaintheoptimizationresultci,asthatinl1-optimizationandNLSSSC.Fortunately,intermsofSVDandthestructureofD(Figs. 1and2),wehave

(11)

whereIdenotesthem-orderedidentitymatrix.Σ′isthefirstnrowsofΣ,V′consistsofthefirstnrowsandthefirstncolumnsofV,andm≫n.Asaresult,whenconstructingthedictionaryVinEq. (10),onlythefirstnrowsandfirstncolumnsofVmustbeprepared,whereastheremainingpartsofVarenotconsideredtomakeanycontributiontothetargettemplatesT.Thus,thelargescaleoptimizationinEq. (10)canbereducedtoamuchsmalleroneasfollows:

(12)

2.3Original and improved ELSSC-tracker

Basedontheabovealgorithm,ourtrackercanbeobtainedwiththeframeworkoftheoriginall1-tracker[8-9](Table2).Weneedtoiterativelysolvethelarge-scalel1-optimizationprobleminEq. (10)twice,uptothreetimesforeachcandidateinthealgorithm,andmorethanvetimesinNLSSST.Theinitialsparsecoecientsc0areconsideredasall-zerovectorsanditerativelysolvetheproblemwithoutanyl1-optimizationissues,asinTable1in[11].Nevertheless,wendthat,inNLSSST,itismoreeectiveandaccuratetoinitializec0asthesolutionofthel1-optimizationproblem.Therefore,thecomputationcomplexityofourtrackerisofthesameorderofmagnitudeasthatofthel1-trackerandNLSSST.Whenweresizealln = 10targetsandN = 200candidateregionsto40 × 40,i.e., m = 1 600 (Figs. 1and2),thentheover-completedictionaryDis1 600 × 3 210andtheorthogonalmatrixVis3 210 × 3 210inEq. (10).Itisverydifficulttosolvethecorrespondingl1-optimizationproblemwithsuchaD(inl1-trackerandNLSSST)orV(inourELSST).

WiththeimprovedELSSC,Σ′isthefirsttenrowsofΣ,andV′consistsofthefirsttenrowsandfirsttencolumnsofV.Thus,eachiterationofeachcandidateregioninELSSTcanbereducedfromthelarge-scalel1-optimizationproblemtoamuchsmalleronebecauseofthemuchsmallerscaleV′∈R10×10.Toovercometheproblemofocclusionsintracking,theanalogoustrivialtemplatesareusedtoconstructthenewdictionaryV″∈R10×30,i.e.,aten-orderedidentitymatrixandten-orderednegativeidentitymatrix.

3Experiments

3.1Experimental setting

Inordertoevaluatetheproposedtracker,experimentson12videosequenceswereconducted,includingSurfer,Dudek,Faceocc2,Animal,Girl,Stone,Car,Cup,Face,Juice,Singer,Sunshade,Bike,CarDark,andJumping[17-19].Thesesequencescoveredalmostallchallengesintracking,includingocclusion(evenheavyocclusion),motionblur,rotation,scalevariation,illuminationvariation,andcomplexbackground.Forcomparison,weusedfourstate-of-the-artalgorithmswiththesameinitialpositionsandthesamerepresentationsofthetargets.Theyweretheincrementallearning-basedtracker(IVT,acommondiscriminanttracker)[14],thecovariance-basedtracker(CovTrack,agenerativetrackeronLie-group)[15],thel1-tracker(agenerativetrackingmethod)[8-9],andtheNLSSST[11].Alltheexperimentswererunonacomputerwitha2.67GHzCPUanda2GBmemory.

Themainparametersusedinourexperimentsaresetasfollows:thenumberofcandidateregionsN=200,thenumberoftemplateregionsisn = 10,andthecandidatesandtargetsareresizedto40×40.

3.2Experimental results for sparsity and stability

ThesparsecoecientsofCR1,…, CR6generatedwiththel1-,theNLSSSC-,theoriginalELSSC-,andtheimprovedELSSC-optimizationareplottedinFig. 3.Inparticular,sixsimilarregionshaveverydierentrepresentationcoecients,whenusingtheoriginall1-optimizationproblem,whichignoresthestructureinformationbetweenregions.Theresultsoftheotherthreealgorithmsaremuchmorestable,becauseofpreservationofthestructuralinformation.Iftworegionsaresimilartoeachother,theyalsohavesimilarsparsecoecients.Thisimprovestherobustnessoftracking;otherwise,thetrackermaydegenerateorevenfailtotrack. CR4forexample,withl1-optimization,canberepresentedbyT2, T8, T6, T7,andT1,andthetrackermayfailtotrackthetopofthebook.Meanwhile,experimentalresultsshowthat,NLSSSCandourtwoELSSCaresparserthantheoriginall1-optimizationproblem.

Fig. 3　Comparisions of sparsity and stability with the original l1-, NLSSSC-, and our ELSSC-optimization. The sparse coefficients only are accurated to the second decimal place.

3.3Experimental results for visual target tracking

Weevaluatetheinvestigatedalgorithmscomparatively,usingthecenterlocationerrors,theaveragesuccessrates,andtheaverageframespersecond.TheresultsareshowninFigs. 4&5andinTables3&4.ThetemplatesofNLSSST,theoriginalELSST,andtheimprovedELSSTareshowninFig. 4(g-o).Overall,ouroriginalandimprovedtrackersoutperformtheotherstate-of-the-artalgorithms.

Forocclusion,vealgorithms,exceptIVT,functionsatisfactorily,especiallyat#206, #366oftheDudeksequenceinFig. 4 (b) (theheadintrackingiscoveredbythehandandglasses), #143, #265, #496oftheFaceocc2sequenceinFig. 4 (c) (theheadintrackingiscoveredbythebook), #85, #108, #433oftheGirlsequenceinFig.4 (e) (theheadintrackingturnsright,turnsback,andblockssomeoneelse),and#56, #104, #301oftheFacesequenceinFig. 4(i) (theheadintrackingisalsocoveredbythebook).Afterthetargetrecoversfromocclusion,thesevetrackerscanseekitquickly.IVTworkspoorly,evenlosesthetargetin#10oftheGirlsequence(Fig. 5(e)),becausethenumberofpositiveandnegativesamplesislimited(consideringthelearningeciency),andtheincrementalupdatingoftheclassierinIVTislesseective.CovTrackinghasalargesizeofcandidates(basedonthedenitionofintegralimage,thefeatureextractionofthesecandidatesissofast,thatitscostcanbeignored),whichmakesitrobustforocclusion,scalevariation,andblur.NLSSSTandouroriginalandimprovedtrackersallworkwell,whenthetargetsareoccluded;ourtwotrackersworkevenbetter.

Formotionblur,ourtwotrackersworkbetterthanIVTandtheoriginall1-tracker.Moreover,CovTrackingalsorevealsitsabilitytohandleblur(e.g., #4, #9,and#38inFig. 4(d,o).Intheformersequence,theanimalrunsandjumpsfast(motionblur)withalotofwatersplashing(occlusion),whileinthelatter,themanropesskippingandthecameracannottaketheclearfaceoftheman.IVTandl1-trackerfailbothfrom#4inFig. 4(d),andneverrecoverafterthat.OuroriginalandimprovedELSSlostthetargetin#31and#41,thenrecoveredin#33and#44 (Fig. 4(d)).In#12to#21and#44to#71,theimprovedELSSTworksbetterthanoriginalELSST,CovTracking,l1-tracker,andNLSSST.

Forrotationandscalevariation,ourtrackersalsoperformrobustly(Figs. 4(a,c,e,g,j)and5(a,c,e,g,j).Whenthesurferfallsforwardandbackward,thegirlturnsleftandright,movestowardsandawayfromthecamera,themanturnsleftandright,thecarturnsover,andthejuicebottlebecomesbiggerandsmallerinSurfer,Girl,Faceocc2,Car,andJuicesequence,respectively,vetrackersexceptIVTperformwell,especiallytheNLSSS-trackerandourtwoELSSC-trackers.

Inacomplexbackgroundandwithhighilluminationvariance(Fig. 4(f)),therearemanysimilarstonestotrack.Thel1-trackerandourtwotrackersworkbetterthanotherthreetrackers.Cov-trackerfails,becauseitextractsedgeinformationoftargetsasonedimensionoffeatures,andinthissequences,edgeoftargetsareambiguousandhardtobedistinct.SimilarresultsareobtainedfromFig. 4(h,l,m).

Table3summarizestheaveragesuccessrates.GiventhetrackingresultsRTandtheground-truthRG,weusethedetectioncriterioninthePASCALVOCchallenge[16],i.e.,

toevaluatethesuccessrate.Ingeneral,fromtheaboveanalysis,wendthatouroriginalandimprovedELSSC-trackersperformalmostthesame,andtheformerisslightlybetter,especiallyintheDudek,Faceocc2,Surfer,Stone,CarDark,andJumpingsequences(Fig. 5(a,b,c,f,n,o).However,wealsondfromTable4,whichsummarizestheaverageframespersecond,thattheimprovedELSSTworksmuchfasterthantheoriginalELSSTandalmostalltheothertrackers;IVTisfasterthantheimprovedELSSTwhendealingwithSurferandDudeksequences,butitssuccessrateismuchworsethanthatoftheimprovedELSST.Itissensitiveunderthephenomenaofocclusion,rotation,andtargetmotionblur.Theoriginall1-trackerperformswellinmostframes,butitisalsotime-consumingandfailstotracksometimes;Cov-Trackingissuitableforocclusionandrotation,butfailswhenfacingacomplexbackground.

Fig. 4　Some tracking results

Fig. 5　Quantitative evaluation in terms of center location error (in pixel)

VideoIVTCovTrackl1-trackerNLSSSTELSST1ELSST2Sufer0.05150.47700.03880.46460.46670.4052Dudek0.20110.42160.62150.65280.67260.6604Faceocc20.45530.39180.60840.45790.57470.4641Animal0.02180.27010.03360.36920.40780.4117Girl0.02280.21710.48690.48530.40060.4693Stone0.09740.11140.58340.41090.66110.6572Car0.06070.18580.09560.34180.32780.3825Cup0.63000.37690.55980.57380.52380.5637Face0.33410.28060.04790.52480.54960.5827Juice0.07430.42180.51110.52990.51860.5835Singer0.33260.13610.11840.57900.47810.5651Sunshade0.04810.18030.52570.53480.47430.4948Bike0.05760.37210.04510.44380.36080.3917CarDark0.08310.30870.07900.01100.42080.3737Jumping0.05770.27550.07110.08470.45300.4505

Thebesttworesultsareshowninbold.Ouroriginalandimprovedalgorithmsareshowninthelasttwocolumns,respectively.

Table 4　Average Frames per Second

Thebesttworesultsareshowninbold.Ouroriginalandimprovedalgorithmsareshowninthelasttwocolumns,respectively.

4Conclusions

Inthisstudy,todealwithsparsityandinstabilityinthel1-optimizationproblem[10-12]andthehightimecomplexityoftheNLSSSC-tracker[11],weproposeanovelefficienttracker,i.e.,theEuclideanlocal-structureconstraintbasedsparsecoding(ELSSC).Ournewalgorithmisal1-trackerwithareconstructedover-completedictionary,whichisdierentfromthatintheoriginall1-trackerandNLSSSC-tracker.Moreover,wesimplifythelarge-scalel1-optimizationprobleminourtrackertoamuchsmalleroneinourimprovedELSSC-tracker.

Comparedwiththeoriginall1-tracker,ourELSSC-trackerintroducesthestructureinformationamongthecandidateregionsgeneratedbytheBayesianinferencetothel1-tracker,similartothatintheNLSSSC-tracker.Withourderivation,theoptimizationprocedureofourtracker(Eq.(10))canbesolvedasthatinthel1-optimizationbutverydierentlyfromthatintheNLSSSC.Furthermore,ourimprovedtrackerismuchmoreecientthanthel1-trackerandNLSSSC-tracker.Ourexperimentsdemonstratethesparsity,stability,andeciencyofourtracker.

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Authorintroduction

HongyuanWANG,male,wasbornin1960,ProfessorofChangzhouUniversity.Hisresearchinterestisimageprocessingandrecognition,artificialintelligence.Hehaspublishedover20papersininternationaljournalsandconferences.

JiZHANG,male,wasbornin1981,LecturerofChangzhouUniversity.Hisresearchinterestisimageprocessingandrecognition.Hehaspublishedfivepapersininternationaljournalsandconferences.

FuhuaCHEN,male,wasbornin1966,AssistantProfessorofWestLibertyUniversity.Hisresearchinterestisvariationimagesegmentationandinverseproblems.Hiscurrentresearchalsoinvolvesobjecttrackingandpersonre-identification.HehaspublishedovertenpapersininternationaljournalscitedbySCIorEI.

DOI:10.11992/tis.201507073

Received Date：2015-07-31. Online Pulication：2015-09-30.

Foundation Item：National Natural Foundation of China under Grant (61572085,61502058).

Corresponding Author：Hongyuan Wang. E-mail: hywang@cczu.edu.cn.

CLC Number：TP18; TP301.6

Document Code：AArticle ID：1673-4785(2016)01-0136-12

网络出版地址：http://www.cnki.net/kcms/detail/23.1538.tp.201509030.1456.002.html