Measurement-to-Track Association for Nontraditional Measurements

Ronald Mahler Unified Data Fusion Sciences, Inc. Eagan, MN, 55122, USA. USDFRPSM@gmail.com Abstract Data fusion algorithms must typically address notonlykinematicissuesthatis,targettrackingbut alsononkinematicsforexample,targetidentification, threatestimation,intentassessment,etc.Whereas kinematicsinvolvestraditionalmeasurementssuchas radardetections,nonkinematicstypicallyinvolvesnon-traditionalmeasurementssuchasquantizeddata, attributes,features,natural-languagestatements,and inference rules.The kinematic vs. nonkinematic chasm isoftenbridgedbygraftingsomeexpert-system approach(fuzzylogic,Dempster-Shafer,rule-based inference)intoasingle-ormulti-hypothesismultitarget tracking algorithm, using ad hoc methods.The purpose of this paper is to show thatconventional measurement-to-trackassociationtheorycanbedirectlyextendedto nontraditionalmeasurementsinaBayesianmanner.Conceptssuchasassociationlikelihood,association distance,hypothesisprobability,andglobalnearest-neighbor distance are defined, and explicit formulas are derived for specific kinds of nontraditional evidence. Keywords:Dataassociation,measurement-to-track association,non-traditionalmeasurements,randomsets, generalized likelihood function. 1Introduction Recentyearshaveseentheemergenceofmultitarget detectionandtrackingalgorithmsthatavoidexplicit measurement-to-trackassociation(MTA).These algorithmsincludeacompleteBayesianformulationof theproblem:thegeneralmultitargetBayesfilter[3, Chapter14],alongwithitsapproximations,thePHD, CPHD,andmulti-Bernoullifilters[3,Chapters16,17].UsingthetechniquesofChapters3-6of[3],thesefilters alsoprovideacompleteBayesianformulationfor processingnontraditionalmeasurementtypessuchas quantizeddata,attributes,features,natural-language statements, and even inference rules (see Section 14.4.2 of [3]). This complete Bayesian formulation is not, however, thesubjectofthispaper.Thedominantmultitarget tracking methodologyfor both legacy algorithms and for mostongoingdatafusionalgorithmR&Disnotthe MTA-freeparadigmjustmentioned,butratherMTA itself.Thusonemightask:Isitpossibletoadaptthe completeBayesianformulationsothatit,oratleastits majoraspects,canbeincorporatedintolegacyMTA-basedalgorithms?Iclaimthatthisquestioncanbe answeredintheaffirmative,buttoansweritonemust address a major gap in MTA theory. AlthoughMTAhasbeenappliedtoarangeof informationfusionproblems,itstheoreticalfoundations (seeSection3andChapter10of[3])remainthoseof multitargettrackingand,specifically,thoseassociated withaparticularkindofinputdata:kinematicmeasurements.MTAaroseasawayofextendingthe purview of the Kalman filter to the multitarget realm. In MTA, a set of measurement-vectorspositions or bearingangles,forexampleiscollected.Thenthe followingquestionisposed:Whichtargettracks generatedwhichofthesemeasurements,andwhich measurements cannot be attributed to any track?Various techniquesgating,nearest-neighbor,globalnearest-neighbor,etc.areusedtodeterminethebestpossible association.In this case MTA leads to what is known as a single-hypothesis tracker (SHT). SHTs tend to exhibit non-robust performance.Thus, moregenerally,onecanpropagateatableofsuboptimal associations,alongwiththeirprobabilitiesofbeingtrue.InthiscaseMTAleadstothecurrentworkhorseof practicalmultitargetdetectionandtracking,themulti-hypothesis tracker (MHT). Alloftheseapproachesdependontheabilityto definewhatitmeansforameasurementtobeneara trackandthustohavebeenplausiblygeneratedbythat track.Thatis,theyalldependontheabilitytodefine distance between measurements and tracks. The theoretical foundations of MTA are based on the presumptionthatcollectedmeasurements,liketracks,are kinematicentitiesthatcanbeeasilyrepresentedin Gaussiantermsi.e.,intermsofvectorsandcovariance matrices.Inthiscase,itiseasytodefinetheconceptof an association distance.Most commonly, this takes the formofaMahalanobisdistanceanditsmultitarget generalizations. Butwhatifthemeasurementsarenotkinematictheypertain,forexample,totargetidentity?Typical examplesofsuchnontraditionalmeasurementsare 14th International Conference on Information FusionChicago, Illinois, USA, July 5-8, 2011978-0-9824438-3-5 2011 ISIF 1454 attributes extracted by human operators, features extracted bydigitalsignalprocessing(DSP)algorithms,natural-languagestatements,andinferencerules.Whatdoes distance between tracks and measurements mean in this case? Themosttypicalapproachestothisproblemhaveconsistedofbottom-up,adhocintegrationsoffamiliar expertsystemsmethodsfuzzylogic,Dempster-Shafer theory,etc.intoMTA-basedalgorithmssuchasMHT.(See[4]forsummariesofsomeofthesetechniques.)It remainsthecase,however,thatsuchmethodsremain controversial,especiallyamongBayesians,whooften describe them as inherently heuristic or worse. Thepurposeofthispaperistoextendstandard kinematicMTAtheorytonontraditionalmeasurementsin asystematic,top-down,theoreticallydisciplined,andas-Bayesian-as-possiblemanner.Iamnotawareofany previousworkthathasevenattemptedtodothiswhich consequentiallymeansthatIknowofnomeaningful precedentsintheliterature.(Ifreadersknowofsuch prededents, I would be happy to learn of them.) My approach is to review traditional kinematic MTA theory(Section3),summarizemyBayesiantheoryof generalizedmeasurementsandgeneralizedlikelihood functions (Section 4), and then show how to integrate the two (Sections 5, 6).Specifically, the systematic treatment ofconventionalMTAinSection3is,inSection5, repeatedessentiallyverbatimexceptthatgeneralized likelihoods(fornontraditionalmeasurements)are substitutedinplaceofconventionallikelihoods(for conventionalmeasurements).Whatresultsisan conceptuallyparsimoniousandstraightforward generalizationofconventionalMTAtheoryto nontraditional measurements. ThusIwilldefineconceptssuchasassociation likelihood,associationdistance,hypothesisprobability, andglobalnearest-neighbordistancefornon-traditional measurements.I will derive explicit formulas for specific kindsofnontraditionalmeasurements.Specialattention willbedevotedtothecasewhenprobabilityofdetection isunity,sincerelativelysimpleclosed-formformulascan be derived under this assumption.The main results of the paperaretheformulasforassociationlikelihoodand hypothesisprobability,Eqs.(49,54),andtheexplicit closed-form formulas for generalized likelihoods for fuzzy and fuzzy Dempster-Shafer measurements in Section 6.1 Butitisnecessarytoalsopointoutwhatthispaper does not do.First, my viewpoint is unreservedly Bayesian.Assuch,thisisnotthepropervenueforasurveyor assessmentofproposedapplicationsofDempster-Shafer, fuzzylogic,DSmT,etc.,tomultitargettracking.Second, I am not proposing a new formulation ofMTA.Rather, I amproposinganextensionofexistingMTAtheoryto nontraditional data.Third, my purpose is to report a new theoreticaladvance,nottoengageintutorialexpositions 1WhiletheresultofEq.(87-89)mightseemlikethemainresultofthe paper,inactualityitisanear-triviality.Itwouldhavelittlevalue without the existence of the actual main results. of existing results.The Bayesian theory of non-traditional measurementsdevelopedin[3]issummarizedinSection 4.I present brief examples of the process of representing nontraditionalmeasurementsasrandomsetsandthen specifyingtheirlikelihoodfunctions(e.g.,Eqs.(30,31)).But for more concrete examples, the reader should consult Chapter3of[3].Fourth,mypurposeistoproposean inherentlymathematicallycomplextheoreticaladvance.Whileimplementationandsimulationareappropriate subjectsforfutureresearch,thereaderwilldiscoverthat theentirepagelimitisrequiredjusttosystematically elucidatethetheoreticalapproach.2Fifthandfinally,the purposeofthispaperisnottoprovideanoverview.The approachrequiressystematicadherencetoa mathematicallypreciseBayesianmethodologywiththe aimofderivingexplicitclosed-formformulasthatcanbe used in practice for different kinds of non-traditional data.Assuch,theexpositionisnomoreandnoless mathematicalthanitmustbeifvalidresultsaretobe reported in sufficient detail for the engaged (as opposed to cursory) reader. Thepaperisorganizedasfollows.Itexpandsupon conceptsbrieflyintroducedinSection10.8,pp.341-342, of[3].Single-andmulti-hypothesistrackersarebriefly reviewed in Section 2.The basic theory of measurement-to-track association is reviewed in Section 3.Generalized measurementsandgeneralizedlikelihoodfunctionsare brieflyreviewedinSection4.Section5describesthe extensionofthematerialinSection3tosuch measurements.InSection6,thistheoryisappliedto arrive at closed-form formulas under certain assumptions.Section 7 describes how to extend it to joint kinematic and nonkinematic processing.Conclusions are in Section 8. 2Single- and multi-hypothesis trackers Probablythetwomostcommonmultitargettracking algorithmsinpracticalapplicationarethesingle-hypothesis tracker (SHT) and the multi-hypothesis tracker (MHT).AMHTrecursivelypropagatestwoitems throughtime:atracktableTk|kandahypothesistableHk|k,usingatime-updatefollowedbyameasurement-update: ! Tk|k,Hk|k ! Tk+1|k,Hk+1|k !Tk+1|k+1,Hk+1|k+1 ! Eachtrack!isamodelofapossiblereal-worldtarget, based on accumulated evidence.It has the form! = (x,P)wherexistheestimatedtargetstateandPthe associated error-covariance matrix.Each hypothesis!in 2Imightadd:ThefoundingIEEEPHDandCPHDfilterpapers[5,6] containednoimplementationsorsimulations.Despitetheseheresies, progress in information fusion does not appear to have suffered. 1455 Hk|kisanassociationi.e.,anassumptionabouthow anygiventargetgeneratedanygivenmeasurementif indeedaparticulartargetgeneratedanymeasurement,or ifaparticularmeasurementwasgeneratedbyanytarget.It is assumed that !="" 1 ) (|k kp. (1) In terms of the most typical implementations seen in theliterature,anMHTconsistsofabankofextended Kalmanfilters(EKFs)orunscentedKalmanfilters (UKFs).Givenanyparticularassociation!,the measurements are used to data-update the predicted tracks withwhichtheyareassociated.Thuseachassociation correspondstoaparticularsetofupdatedtracksthatis, to some subset of the track table.In this sense, each!is a model of what ground truth looks like, andpk+1|k+1(!)istheprobabilitythatthismodelaccuratelyrepresents ground truth.Itispossibletoavoidthemulti-hypothesisstructure bypropagatingonlythehighest-probabilityhypothesis ratherthantheentirehypothesistable.Thisis accomplishedbyusingaglobalassociationdistanceto determine the optimal measurement-to-track association at anygiventime-step.Associationdistancesaredefined from the formulas for hypothesis probabilities.The result, various kinds of SHTs, are computationally more tractable but also typically exhibit poorer tracking performance. ThebasictheoryisintroducedinSection10.3.1of [3] and is reviewed in Section 3.3Measurement-to-track association (review)ThissectiondrawsheavilyfromSection10.3.1of[3].Suppose that, at time-stepk+1, X ={(x1,P1),,(xn,Pn)}(2) isthesetoftrackspredictedfromtheprevioustime-step, wherex1,,xnaretheirstatesandP1,,Pnaretheir error-covariancematrices.Thetrackdensity corresponding to(xi,Pi)is the Gaussian distribution ) ( ) | (i PiN i f x x x ! =.(3) Also, let the sensor likelihood function at timek+1be ) ( ) | (i R iH N f x z x z ! =(4) and let the measurement-set collected at time-stepk+1be Z ={z1,,zm}(5) with|Z| = m. 3.1Definition of an associationA measurement-to-track association is a function !: {1,,n}!{0,1,,m} (6) which has the following property: !(i) = !(i") > 0 implies thati=i".Thisfunctionhasthefollowingintuitive interpretation: (a)foreveryi=1,,n,if!(i)>0thenthe measurementz!(i)isuniquelyassociatedwiththe trackxi ; (b) if!(i) = 0then no measurement is associated withxi

(i.e., the track was not detected); and(c) in either case, those measurements in Zwhich are not equal to anyz!(i)with!(i) > 0are false detections. Atoneextreme,if!(i)=0forallithennotarget generated a measurement and so every measurement inZmustbeafalsedetection.Inthiscase,abbreviate!=0and adopt the convention d! = 0.At the other extreme, if !(i)>0forallitheneverytrackgenerateda measurement.3.2Local association distanceSupposeforthemomentthatasinglekinematic measurementzhasbeencollected.Wearetoestimate whichofthepredictedtracks(x1,P1),,(xn,Pn)most likelygeneratedit,assumingthatitwasnotafalse detection.Thetotallikelihoodthatzwasgeneratedby(xi,Pi)i.e., the association likelihoodis !" = x x x z z d i f f i ) | ( ) | ( ) | ( !(7) !" # " = x x x x z d N H Ni P Ri) ( ) ((8) ) (iH HP RH NTix z ! =+ (9) ( )221) | ( exp) ( 2 det1iTidH HP Rx z ! "+=# (10) whered(z|xi)2=(z-Hxi)T(R+HPiHT)#1(z-Hxi) .(11) 3.3Global association distanceSupposethatanentiremeasurement-setZ={z1,,zm}with|Z|=mhasbeencollected.Assumethatthefalse alarmprocessisPoissonwithclutterrate"andspatial distributionc(z),whereIamsuppressingthetime-indexkfor notational clarity. Case1:pD=1.Beginbyassumingthatprobability of detection is unity, in which case an association is a one-to-one function#:{1,,n}!{1,,m}. (12) LetW#$Zbe the subset of detectionsthat is, thosezthat equalz#(i)for somei = 1,,n. Giventhis,theglobalassociationlikelihoodthat is, the likelihood that# matches the measurements with the tracks, taking false alarms into accountis, from Eqs. (10.58) and (10.94) of [3], !=" =niii c X Z1) () | ( ) ( ) | (# ## z ! ! (13)where ) ( ) ( zzc e cW eZ! ""!#$$=.(14) Thevalueoftheglobalassociationlikelihoodtendstobe largerforthose#whichmoreplausiblyassociate measurements to predicted tracks. Letp#,0bethepriorprobabilityoftheassociation#.Sincethereisnoapriorireasontopreferone associationoveranother,p#,0isauniformdistribution.1456 Thustheposteriorprobabilitythattheassociation#is the correct one is ) | () | () | (''X ZX ZX Zp!"!!!!!!# =$. (15) The optimal association is given by the MAP estimate ) | ( max arg max arg X Z p! ! ! "! ! = =. (16) Theglobalassociationdistanceforanassociation#is defined from Eq. (15) by ) | ( log 22X Z d! !! " =.(17) Thus the optimal association is the one that minimizes this association distance. FromEq.(10.91)of[3],wefindthatthespecific formula for Eq. (15) is ) ) | ( exp( ) (221X Z d c p! !! " # $ (18) where!="" + " =nii iTiTi iH H HP R H X Z d1) (1) (2) ( ) ( ) ( ) | ( x z x z# # # (19)Assumethatc(z)=cisuniformoversomeregionof space.Thenthequantityc(#)=e#"("c)m#nisnolonger functionally dependent on#and thus ) ) | ( exp(221X Z d p! !" #. (20) Thus the global association distance can be taken to be as in Eq. (19):) | ( X Z d d! ! =.(21) Thisisthedefinitionmostcommonlyemployedindata association algorithms.Case 2: pD < 1.Now consider the more general case, inwhichtheprobabilityofdetectionisconstantbutnot unity,andthus!isanarbitraryassociation.Thenthe global association likelihood is (see Eq. (G.238) of [3]) !="# # " =nii Dn nDi p c p X Z1) () | ( ) ( ) 1 ( ) | ($ %$%z ! ! (22) wheren!denotesthenumberofdetectionsassociated with!.Inthiscasetheformulafortheassociation probabilityp!is (see Eqs. (10.115) and (10.116) of [3]):) ) | ( exp( ) (221X Z d F c K pn! ! !!!" # # # $(23) where! ! n nDnDp p K"" = ) 1 ( (24) ) ( ) (1zzc e cW eZk! "#!$%%+=(25) !> +=0 ) ( ) ( 2 det1i iLTiH HP RF""#(26) !>"" + " =0 ) ( :) (1) (2) ( ) ( ) ( ) | (i ii ITiTi iH H HP R H X Z d#$ $ #x z x z

(27) Eveniftheclutterspatialdistributionisuniform, associationdistanceismuchmorecomplexthanwasthe case for unity probability of detection.4Nontraditional measurements (review) InChapter5of[3],Iintroducedtheconceptofunambiguouslygeneratedambiguous(UGA) measurementsandtheirgeneralizedlikelihood functions.Nontraditionalmeasurementse.g., attributes,features,natural-languagestatements,and inferencerulesarerepresentedasrandom(closed) subsets%ofameasurementspaceZ0thatisas generalizedmeasurements.Inturn,generalized measurementsaremediatedbygeneralizedlikelihood functions, which are defined as fk+1(%|x)=Pr($k+1(x) & %)(28) wherez=$k+1(x)isadeterministicmeasurement model.Ishowedhowvariousexpert-systemformalisms forrepresentingnontraditionalmeasurementsfuzzy logic,Dempster-Shafertheory,rule-basedinferencecan be used to construct random set models. Forexample,consideranimprecisemeasurement, whichisjusta(closed)subsetSofthemeasurement space. (A quantized measurement is a typical example of animprecisemeasurement.Thistypeofmeasurementis explicitlyconsideredinacompanionpaper[2].)Ifthe imprecisemeasurementisunderstoodasbeing deterministic,then%=Sandthecorresponding generalized likelihood function isfk+1(S|x)=Pr($k+1(x)&S)=1S($k+1(x)) (29) where1S(z)is the set indicator function ofS. Asanotherexample,considerafuzzy measurementi.e.,afuzzymembershipfunctiong(z)onZ0.Suchameasurementcanbeunderstoodasan impreciselyspecifiedimprecisemeasurementmeaningthatthemeasurementisimprecise,butthatthe specificformoftheimprecisionisunclear,havingmany possible formsSa = {z| a'g(z)}for0 ' a ' 1.Using the random set representation (g={z|A ' g(z)}(30) ofg(z),whereAisauniformlydistributedrandom numberontheunitinterval[0,1],Ishowedthatits corresponding generalized likelihood is ([3], Eq. 5.29): fk+1(g|x)= g($k+1(x)) . (31) Asathirdexample,considerafuzzyDempster-Shafer(FDS)measurementi.e.,abasicmass assignment (g)on the fuzzy subsetsgofZ0, defined by the properties:(g) ) 0for allg; (g) = 0ifg = 0;(g)=0forallbutafinitenumberofg(thefocal fuzzy sets of); and (g (g) = 1. (When the focal fuzzy sets ofare all crisp, thenis a conventional Dempster-Shafer basic mass assignment.)A FDSmeasurementisafurthergeneralizationofthe conceptofanimprecisemeasurement,inwhichmultiple hypotheses are required to represent the uncertainty in the choiceofimprecisemeasurement.Employingarandom setrepresentation(of,Ishowedthatits corresponding generalized likelihood is ([3], Eq. (5.73)):! + +" =gk kg g f )) ( ( ) ( ) | (1 1x x # .(32)1457 Iderivedgeneralizedlikelihoodfunctionsforother nontraditionalmeasurements,suchasfuzzyinference rulesg*g"([3], Eq. (5.80)): ( ) )) ( ( ' 121)) ( )( ' ( ) | ' (1 1 | 1x x x+ + +! + " = #k k kg g g g g f $ $ (33) using a random set representation (g*g" forg*g". Now,alloftheabovegeneralizedlikelihood functionformulasarebasedonacommonassumption:thatthegeneralizedmeasurementisdeterministic.Even thoughrandomsets(g,(,and(g*g",areusedto representthenontraditionalmeasurementsg,,andg*g",thesemeasurementsarethemselvesnotrandom.That is, they do not arise as specific instantiations of some random variable. (For a more complete discussion see, for example, Eq. (4.23) of [3].) In the companion paper [2], I showed how to extend thisformulationtoincludenontraditionalmeasurements thatareinstantiationsofarandomvariable.Inthiscase, the generalized measurement has the form %# Vk+1.(34)where%istherandomsetmodelofthedeterministic nontraditionalmeasurement,andVk+1isanadditive randomnoisevector.Inthiscasethegeneralized likelihoodofthenontraditionalmeasurement,taking randomness into account, is fk+1(%|x)= Pr($(x) & %#Vk+1). (35) !+ "# = z x z z d fk) | ( ) (1 (36) where%(z)=Pr(z & %)(37) is Goodmans one-point covering function of%(see [3], Eq. (4.20)).Asaspecialcase,let%=%gmodelafuzzy measurementg(z).Then Eq. (35) reduces to!+ +" = z x z z x d f g g fk k) | ( ) ( ) | (1 1.(38) Asa simple closed-form example of Eq. (38), let ) ( ) | (1x z x z H N fR k! =+ (39)) ( 2 det ) ( c z z ! " =CN C g #. (40) Then Eq. (38) becomes) ( 2 det ) | (1x c x H N C g fR C k! " =+ +#. (41) 5Measurement-to-track association (nontraditional measurements)Suppose that, at time-stepk+1, X ={(x1,P1),,(xn,Pn)}(42) isthesetoftrackspredictedfromtheprevioustime-step, wherex1,,xnaretheirstatesandP1,,Pnaretheir error-covariancematrices.Thetrackdensities correspondingtothe(xi,Pi)arenotnecessarilylinear-Gaussiandistributions(thoughthe(xn,Pn)arecomputed from them): ) | ( i f x. (43) ThesetrackdensitiesarisethoughstandardMTAtrack propagation.Ingeneral,thepropagationisaccomplished viathesingle-targetBayesfilterratherthan,say,anEKF or UKF (see Section 5.3).Thesourcelikelihoodfunctionattimek+1is definedongeneralizedmeasurements.Itisthereforea generalizedlikelihoodfunction,andinparticularitwill not necessarily be linear-Gaussian: ) | ( x !if. (44) Assumethatthemeasurement-setcollectedattime-stepk+1consists of nontraditional measurements, Z ={%1,,%m}(45) with|Z| = m. 5.1Local association distance (nontraditional measurements) IproceedasinSection3.2.Supposethatasingle nontraditionalmeasurement%hasbeencollected.We aretoestimatewhichofthepredictedtracks (x1,P1),,(xn,Pn)mostlikelygeneratedit,assumingthat itwasnotafalsedetection.Theassociationlikelihood for%is then!" # = # x x x d i f f i ) | ( ) | ( ) | ( !. (46) 5.2Global association distance (nontraditional measurements) SupposethatanentiresetZ={%1,,%m}of nontraditionalmeasurementswith|Z|=mhasbeen collected.Becausethemeasurementsaregeneralized, theymustbemediatedbyageneralizedlikelihood functionf(%|x).Consequently,theclutterprocessmust bedefinedongeneralizedmeasurements,anditsspatial functionc(%)musthavetheformofageneralized likelihoodfunctionthatis,itmustbeunitless.For example,if%=%grepresentsafuzzymeasurementg(z), one might choose !" = # = z z z d c g c g cg) ( ) ( ) ( ) (.(47) For current purposes and in what follows, however, I will assume that c(%) = cis constant.Case1:pD=1.Giventhis,asinSection3.3begin byassumingthatprobabilityofdetectionisunity,in which case an association is a one-to-one function#:{1,,n}!{1,,m}. (48) LetW#denote the set of %inZ that equal%#(i)for some i = 1,,n. Thentheglobalassociationlikelihood,whichisa generalized likelihood function, is !=" # =niii c X Z1) () | ( ) ( ) | ($ $$ ! ! (49)where n mW eZc c c!!= " = #) ( ) ( ) ( $ $ %%z (50) which does not functionally depend on#.In this case the probability of the association+is!='') | () | ("###X ZX Zp!! (51)1458 !=" #niii1) () | ($!(52) and so !=" # =niIii d1) 21 2) | ( log$ $! (53) can be taken to be a global association distance. Case2:pD