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378Vol.37,No.820118ACTAAUTOMATICASINICAAugust,201111,Dirichlet,.,Dirichlet.;Dirichlet,.,.,.,.,.,,,,,DirichletDOI10.3724/SP.J.1004.2011.00963MultipleTargetTrackingAlgorithmBasedonOnlineEstimationofTargetBirthIntensityYANXiao-Xi1HANChong-Zhao1AbstractAsfarastheunknowntargetbirthintensityinmultipletargettrackingisconcerned,anonlineestimationalgorithmoftargetbirthintensityisproposedtoimprovetheperformanceofprobabilityhypothesisdensity¯lterinmultipletargettracking.The¯nitemixturemodelisadoptedtomodeltheunknownintensityoftargetbirth.Dirichletdistributionwithnegativeexponentparameters,whichonlydependsonthemixingweights,isusedasthepriordistributionofparametersinthemixturemodel.TheonlineestimationformulationofmixingweightisderivedbyLagrangemultiplierinthesenseofmaximumaposterior.TheinstabilityofDirichletdistributionwithnegativeexponentparametersisappliedindrivingthecomponentsirrelevantwithbirthtargetstoextinctionduringtheonlineestimationprocedureofmixingweights.Stochasticapproximationprocedureisregardedasthestrategyofonlineestimationofcomponentmeanandcovariance.Theonlineestimationformulationsofcomponentmeanandcovariancearederivedbasedonmissingdata.Aninitializationmethodisdevelopedbyaddingauniformdistributionintothemixingmodel,undertheconstraintthatnopriordistributionoftargetbirthisobtainedininitialization.Fromthestandpointofcurrentestimatesofmultipletargetstates,themethodofachievingthedataofbirthtargets,whichmakesfulluseoftheabilityofprobabilityhypothesisdensity¯ltertoreducethee®ectofclutter,ispresented.Simulationresultsshowthattheproposedonlineestimationalgorithmoftargetbirthintensitycanimprovetheperformanceofprobabilityhypothesisdensity¯lterinmultipletargettracking.KeywordsMultipletargettracking,probabilityhypothesisdensity(PHD),targetbirthintensity,onlineestimation,maximumaposterior(MAP),Dirichletdistribution.2010-09-292010-12-27ManuscriptreceivedSeptember29,2010;acceptedDecember27,2010(973)(2007CB311006),(60921003)SupportedbyNationalBasicResearchProgramofChina(973Program)(2007CB311006)andFoundationforInnovativeRe-searchGroupsofNationalNaturalScienceFoundationofChina(60921003)1.7100491.KeyLaboratoryforIntelligentNetworksandNetworkSecu-rityofMinistryofEducationandStateKeyLaboratoryforMan-ufacturingSystemsEngineering,InstituteofIntegratedAutoma-tion,SchoolofElectronicandInformationEngineering,Xi0anJiaotongUniversity,Xi0an710049,,[1].,,[2].,,,.,,.,,,.,96437.[3],,(Probabilityhypothesisdensity,PHD)[4¡6].PHD\,;.,PHD,.PHD,,.PHD:(SequentialMonteCarlo,SMC)[7¡9],[10¡11];,;[12¡13].,PHD(CardinalizedPHD,CPHD)PHD,CPHDPHD[14¡16].,,[17¡18].PHDCPHD,,[19¡20][21¡22][23¡24][25¡27][28][29][30].PHD,,.,.,,.PHD:,,.:PHDSMC,,,;PHD,.,,.,Dirichlet,PHD.,.Dirichlet.,,,,PHD.1Bayesfkjk¡1(XjZ1:k¡1)=Zfkjk¡1(XjW)£fk¡1jk¡1(WjZ1:k¡1)±W(1)fkjk(XjZ1:k)=K¡1fkjk¡1(ZkjX)£fkjk¡1(XjZ1:k¡1)(2),X,Zkk,Z1:k1k,fkjk(XjZ1:k),fkjk¡1(XjZ1:k¡1),fkjk¡1(XjW),Gkjk¡1(ZkjX),K=Rfkjk¡1(ZkjX)fkjk¡1(XjZ1:k¡1)±XBayes.,PHDDkjk¡1(xkjZ1:k¡1)=°k(xk)+Z[¯kjk¡1(xkjxk¡1)+ekjk¡1(xk¡1)fkjk¡1(xkjxk¡1)]£Dk¡1jk¡1(xk¡1jZ1:k¡1)dxk¡1(3)Dkjk(xkjZ1:k)=(1¡PD(xk))Dkjk¡1(xkjZ1:k¡1)+Xz2Zk'k;z(xk)Dk(z)Dkjk¡1(xkjZ1:k¡1)(4),D(¢),°k(xk),¯kjk¡1(xkjxk¡1)xk¡1,Dk(z)=·k(z)+Ck(z)8:965z,·k(z),ekjk¡1(xk),fkjk¡1(xkjxk¡1),'k;z(xk)=PD(xk)gkjk(zjxk),Ck(z)=R'k;z(xk)Dkjk¡1(xkjZ1:k¡1)dx,gkjk(zjxk),PD(xk)[4;6].PHDSMC:,SMCPHD[7].PHD,,PHD.,1,PHD1.PHDSMC:1.k¡1~x(i)k»qk³¢jx(i)k¡1;Zk´,~w(i)kjk¡1=Ákjk¡1³~x(i)k;x(i)k¡1´qk³~x(i)kjx(i)k¡1;Zk´w(i)k¡1(5),i=1;¢¢¢;Lk¡1,Lk¡1k¡1,Á(xk;xk¡1)=ekjk¡1(xk¡1)fkjk¡1£(xkjxk¡1)+¯(xkjxk¡1).k~x(i)k»pk(¢jZk)~w(i)kjk¡1=1Jk°k³~x(i)k´pk³~x(i)kjZk´(6),i=Lk¡1+1;¢¢¢;Lk¡1+Jk,Jk.2.kz2Zk,Ck(z)=Lk¡1+JkXj=1'k;z³~x(j)k´~w(j)kjk¡1(7),1~w(i)k=³1¡PD³~x(i)k´´~w(i)kjk¡1+Xz2Zk'k;z³~x(i)k´·k(z)+Ck(z)~w(i)kjk¡1(8),i=1;¢¢¢;Lk¡1+Jk.3.,^Nkjk=PLk¡1+Jkj=1~w(j)k;,n³~w(i)k=^Nkjk´;~x(i)kon³w(i)k=^Nkjk´;x(i)koLki=1;,^NkjkkPHDnw(i)k;x(i)koLki=1.PHD,^Nkjkk.°k(xk)(3)(6)k,..°(x),,PHD.Dirichlet.2,.,,,°(x)=´p(x)(9),´,p(x),p(x).k´k,1k¡1´k^´k^´k=^´1+¢¢¢+^´k¡1k¡1(10),k.,,,[31]p(xjµ)=MXm=1¼mp(xjµm)(11),M,µm=f¹m;Ámg¹mÁmm,µ=fµ1;¢¢¢;µM;¼1;¢¢¢;¼Mg96637,MXm=1¼m=1(12)nX(n)=©x(1);¢¢¢;x(n)ª(11)p(xjµ)logp¡X(n)jµ¢=nXi=1logMXm=1¼mp¡x(i)jµm¢(13),(),.,,.°(x)´¼m¹mÁm.Dirichlet¼m;,¹mÁm;,.2.1DirichletDir(®®®)®®®=(®1;¢¢¢;®M),,®mm[32].Dirichlet(12),Dirichlet,p(¼1;¢¢¢;¼M;®®®)=1B(®®®)MYm=1¼®m¡1m(14),B(®®®)¯B(®®®)=MQm=1¡(®m)¡µMPm=1®m¶(15)m¼mE(¼m)=®m®0(16),®0=PMm=1®m.ijcov(¼i;¼j)=¡®i®j®20(®0+1)(17),.(11)µDirichletp(µ)/expÃcmMXm=1log¼m!(18),cm=¡N=2m,N[33].µp(µ)¼¼¼;cm=¡N=2m.Dirichlet,.(Maximumapos-terior,MAP)µ.X(n)=©x(1);¢¢¢;x(n)ª,µMAP^µ=argmaxµ©logp¡X(n)jµ¢+logp(µ)ª(19)MAP,p¡µjX(n)¢¼m@@¼m¡logp¡X(n)jµ¢+logp(µ)¢=0(20)(12)(20)@@¼mÃlogp¡µjX(n)¢+¸ÃMXm=1¼m¡1!!=0(21),¸.,¸,¼mMAP¼(n)m=1nnXi=1!(n)m¡x(i)¢¡N2n1¡MN2n(22),!(n)m(x)xm!(n)m(x)=¼(n)mp¡xjµ(n)m¢MPm=1¼(n)mp³xjµ(n)m´(23),n+1X(n+1)=©X(n);x(n+1)ª,¼mMAP8:967¼(n+1)m=1n+1n+1Xi=1!(n+1)m¡x(i)¢¡N2(n+1)1¡MN2(n+1)(24)!(n+1)m(x)=¼(n+1)mp¡xjµ(n+1)m¢MPm=1¼(n+1)mp³xjµ(n+1)m´(25)(23)!(n)m(x)(25)!(n+1)m(x),x(n+1).,,,X(n+1)=©X(n);x(n+1)ª!(n+1)m(x)X(n)=©x(1);¢¢¢;x(n)ª!(n)m(x)!(n+1)m(x)¼!(n)m(x)(26),,.,,N=2(n+1)¼N=(2n)=N=(2c)Diri