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RandomSet/PointProcessinMulti-TargetTrackingBa-NguVoEEEDepartmentUniversityofMelbourneAustralia(innoparticularorder):MahlerR.,Singh.S.,DoucetA.,Ma.W.K.,PantaK.,ClarkD.,VoB.T.,CantoniA.,PashaA.,TuanH.D.,BaddeleyA.,ZuyevS.,SchumacherD.TheBayes(single-target)filterMulti-targettrackingSystemrepresentationRandomfiniteset&BayesianMulti-targetfilteringTractablemulti-targetfiltersProbabilityHypothesisDensity(PHD)filterCardinalizedPHDfilterMulti-BernoullifilterConclusionsOutlineTheBayes(single-target)Filterstate-vectortargetmotionstatespaceobservationspacexkxk-1zk-1zkfk|k-1(xk|xk-1)MarkovTransitionDensityMeasurementLikelihoodgk(zk|xk)Objectivemeasurementhistory(z1,…,zk)posterior(filtering)pdfofthestatepk(xk|z1:k)SystemModelstate-vectortargetmotionstatespaceobservationspacexkxk-1zk-1zkBayesfilterpk-1(xk-1|z1:k-1)pk|k-1(xk|z1:k-1)pk(xk|z1:k)predictiondata-updatepk-1(xk-1|z1:k-1)dxk-1fk|k-1(xk|xk-1)gk(zk|xk)K-1pk|k-1(xk|z1:k-1)TheBayes(single-target)Filterpk-1(.|z1:k-1)pk|k-1(.|z1:k-1)pk(.|z1:k)predictiondata-updateBayesfilterN(.;mk-1,Pk-1)N(.;mk|k-1,Pk|k-1)N(.;(mk,Pk)Kalmanfilteri=1N{wk|k-1,xk|k-1}i=1N(i)(i){wk,xk}i=1N(i)(i){wk-1,xk-1}(i)(i)Particlefilterstate-vectortargetmotionstatespaceobservationspacexkxk-1zk-1zkfk|k-1(xk|xk-1)gk(zk|xk)TheBayes(single-target)FilterMulti-targettrackingobservationproducedbytargetstargetmotionstatespaceobservationspace5targets3targetsXk-1XkObjective:JointlyestimatethenumberandstatesoftargetsChallenges:RandomnumberoftargetsandmeasurementsDetectionuncertainty,clutter,associationuncertaintyMulti-targettrackingSystemRepresentation0011X11'00XEstimateiscorrectbutestimationerror???TrueMulti-targetstateEstimatedMulti-targetstate||'||2XXHowcanwemathematicallyrepresentthemulti-targetstate?2targets2targetsUsualpractice:stackindividualstatesintoalargevector!Problem:Remedy:use(')min||'||0permXXX11'00XTrueMulti-targetstate?XEstimatedMulti-targetState2targetsnotarget11'00XTrueMulti-targetstate00XEstimatedMulti-targetState2targets1targetSystemRepresentationWhataretheestimationerrors?Errorbetweenestimateandtruestate(miss-distance)fundamentalinestimation/filtering&controlwell-understoodforsingletarget:Euclideandistance,MSE,etcinthemulti-targetcase:dependsonstaterepresentationFormulti-targetstate:vectorrepresentationdoesn’tadmitmulti-targetmiss-distancefinitesetrepresentationadmitsmulti-targetmiss-distance:distancebetween2finitesetsInfactthe“distance”isadistanceforsetsnotvectors(')min||'||0permXXXSystemRepresentationobservationproducedbytargetstargetmotionstatespaceobservationspace5targets3targetsXk-1XkNumberofmeasurementsandtheirvaluesare(random)variablesOrderingofmeasurementsnotrelevant!Multi-targetmeasurementisrepresentedbyafinitesetSystemRepresentationRFS&BayesianMulti-targetFilteringtargetstargetsetobservedsetXobservationsXZNeedsuitablenotionsofdensity&integrationpk-1(Xk-1|Z1:k-1)pk(Xk|Z1:k)pk|k-1(Xk|Z1:k-1)predictiondata-update1|11:1(|)(|)kkkkkkkKgZXpXZReconceptualizeasageneralizedsingle-targetproblem[Mahler94]Bayesian:Modelstate&observationasRandomFiniteSets[Mahler94]RFS&BayesianMulti-targetFilteringSN(S)=|S|pointprocessorrandomcountingmeasurerandomfinitesetorrandompointpatternstatespaceEstatespaceEBelief“density”off:F(E)[0,)b(T)=Tf(X)dXBelief“distribution”ofb(T)=P(T),TEEProbabilitydensityofp:F(E)[0,)P(T)=Tp(X)m(dX)ProbabilitydistributionofP(T)=P(T),TF(E)F(E)CollectionoffinitesubsetsofEStatespaceMahler’sFiniteSetStatistics(1994)Choquet(1968)TTConventionalintegralSetintegralVoet.al.(2005)PointProcessTheory(1950-1960’s)RFS&BayesianMulti-targetFilteringxx’X’xdeathcreationX’xspawnmotionMulti-targetMotionModelfk|k-1(Xk|Xk-1)Multi-objecttransitiondensityXk=Sk|k-1(Xk-1)Bk|k-1(Xk-1)kEvolutionofeachelementxofagivenmulti-objectstateXk-1Multi-targetObservationModelgk(Zk|Xk)Multi-objectlikelihoodZk=Qk(Xk)Kk(Xk)xzxlikelihoodmisdetectionclutterstatespaceobservationspaceObservationprocessforeachelementxofagivenmulti-objectstateXkpk-1(Xk-1|Z1:k-1)pk(Xk|Z1:k)pk|k-1(Xk|Z1:k-1)predictiondata-updateComputationallyintractableingeneralNoclosedformsolutionParticleorSMCimplementation[Vo,Singh&Doucet03,05,Sidenbladh03,Vihola05,Maetal.06]Restrictedtoaverysmallnumberoftargets)()|()|(1:111|dXZXpXXfskkkkkm)()|()|()|()|(1:11|1:11|dXZXpXZgZXpXZgskkkkkkkkkkkkmMulti-targetBayesFilterMulti-targetBayesfilterParticleMulti-targetBayesFilterAlgorithmfori=1:N,%Initialise=Sample:Compute:end;normaliseweights;fork=1:kmax,fori=1:N,%Update=Sample:Update:end;normaliseweights;resample;MCMCstep;end;()()1:1(|)(),ikNikkkkkXipXZwXd()000~()iXqX()0()00001()(),iNiXipXwXd()()()00000()()iiiwpXqX()()()()()()()1|1111:(|)(|)(|,)iiiiiiikkkkkkkkkkkkkwwgZXfXXqXXZ()()11:~(|,)iikkkkkXqXXZpk-1(Xk-1|Z1:k-1)pk(Xk|Z1:k)pk|k-1(Xk|Z1:k-1)predictiondata-updateMulti-targetBayesfilter:veryexpensive!single-objectBay