Clutter-based

Clutter-based Vo1.34.No.3ACTAAUT0MATICASINICAMarch.2008 Clutter.basedTestStatisticsforAutomatic TrackInitiation KENNEDYHughLachlan AbstractTwoteststatisticsbasedoncluttercharacteristicsarederived.Atentativetrackisconfirmedwhenthetrack-ison— clutterhypothesisisrejected.Aconstantandknownfalsetrackrateresultswhentheassumptionsofthenullhypothesisaretrue. Thefirstteststatisticisbasedontheclutterdensity.Ahighprobabilityoftargetdetectionisresultedwhentheexpecteddistanceto thenearesttargetpeakislessthantheexpecteddistancetothenearestclutterpeak.Thesecondteststatisticisbasedontheclutter amplitude.Ahighprobabilityoftargetdetectionisresultedwhentheexpectedamplitudeofthetargetpeakisgreaterthanthe expectedamplitudeofclutterpeaks.Thebehavioroftheclutter-basedteststatisticsiscomparedwiththetargetvisibilitymethod, usingsimulateddata.Alltrackinitiationmethodsareappliedusingatrackupdaterbasedonprobabilisticdataassociation(PDA), extendedtoincorporatepeakamplitudeinformation,whichisavailable. KeywordsTrackingfilters,targetdetection,automatictrackinitiation,probabilisticdataassociation(PDA) Inoperationalwide-areasurveillancesystems._——forex?? ample,thosedescribedinI1—41一automatictrackiniti— ationisrequiredtoreduceoperatorworkloadandincrease overallsystemeffectiveness.Automatictrackinitiationis theprocesswherebytracksontargetsarepromptlypre- sentedtotheoperatorforassessment.withoutmanualin— tervention.Atentativetrackisstartedonalltarget—like peaksandautomaticallymaintainedbythetrackingsub— system(thetracker).Thesetracksareonlyconfirmedand displayedwhenthetrackconfidenceexceedsanominated upperthreshold;trackswithaconfidencebelowanomi— natedlowerthresholdaredeleted.Theabilityofthetracker todiscriminatebetweentrueandfalsetracksdetermines theaveragefalsetrackrate,theaveragetrackinitiation(or establishment1delay,andtheaveragenumberoftracksheld inthetrackingdatabasefthusthecomputationalload). ProbabilisticdataassociationfPDA)一whichwasini- tilyformulatedin『5],developedandcustomizedinpa- petssuchasI6—7l,andrecentlyreviewedinI8,9】一 iswell-suitedtowideareasurveillancesystemswherethe clutter(orfalsepeak)densityistypicallyhigh,theproba- bilityoftargetdetectionlow,thescan(orrevisit)ratelow, thesurveillancevolumelarge,andthemeasurementnoise high.Intheseenvironments,PDA—basedtrackersprovide robustnessatalowcomputationalcost,alongwithreduced developmentandmaintenancecostsduetotheirrelative simplicity. InI10l,PDAwasextendedtoincorporateautomatic trackinitiation(andtermination),byconsideringthe target—is-not—observableevent.inadditiontotheall- associated..peaks-are-cluttereventoftheoriginalPDAfor—. mulation.Theideahassincebeendevelopedfurther,with observabilityreplacedbyvisibilityt,perceivabilityt11J,and existencel. Collectively.theseapproachesaresometimes referredtoasintegratedPDA(IPDA).InI13I,afixed—lag smootherisusedtorefinetheestimateofexistence;while in[14],theapproachisreformulatedusinganinteracting multiplemode1.Stateestimationandexistenceestimation aretightlycoupledinIPDA—thegainofthetrackingfilter isinfluencedbytheexistenceprobability,whereastheex— istenceprobabilityisdeterminedbythetargetprobability densityrelativetotheclutterdensity.0therPDA—based andnon—PDA—basedtrackinitiationschemesaredescribed in[15—16]and[17—23],respectively ReceivedSeptember13,2007;inrevisedformJanuary7,2008 SupportedbyTechnicalKnockoutSystemsPtyLtd. 1.TaranakiRpad,EdinburghParks,EdinburghSA5111,Australia DOh10.3724/SP.J.1004.2008.00266 Inthispaper,trackinitiationschemesbasedpurelyon clutterstatisticsarepresented.Theproblemisformulated asahypothesistest,withthenullhypothesisbeingthat thetrackisonclutter.Atrackisconfirmedwhenthenull hypothesisisrejected.Whentheassumptionsunderlying thenullhypothesisaretrue,aconstantandknownfalse trackconfirmationprobabilityresults—thesizeofthetest; furthermore.approachingthetrackinitiationproblemin thiswayhasthepotentialtoreducetrackinitiationde- laysontargetswhenthepriorassumedmeasurementnoise andprocessnoisedistributions(asdefinedintheRand Qmatrices)areinappropr1ateorwhenthestateestimates arepoor.Themethodsarealsosuitableforuseinlegacy systemsemployingnon-Bayesiantrackerst一. Threefiltersfortheautomaticinitiationoftracksinclut— teraredescribedinSection1.Afterabriefoverviewofthe targetvisibilityfilter,twoclutter.onlymethodsarepre- sented.Thefirstisbasedonthespatialdistributionfden— sity1oftheclutterpeaks;thesecondisbasedonthesignal strength(amplitude)oftheclutterpeaks.Theparameters oftheclutterpopulationareestimatedandanytrackon peaksequencesthatareunlikelytobeduetoclutterare confirmed.Teststatisticsareevaluatedusingthenearest peaktoassociatewiththetrackoneachupdate,overthe recenthistoryofthetrack.Thepowerofthetests,i.e.,the achievedtruetrackconfirmationprobability,dependson thetargetpeakcharacteristics.Theamplitudetestispow— erfulwhenthemeansignalstrengthoftargetdetectionsis significantlygreaterthanthemeansignalstrengthofthe falsedetections;thedensitytestispowerfulwhentheex- pecteddistancetothenearesttargetpeakislessthanthe expecteddistancetothenearestclutterpeak.Thetrack initiationfiltersareappliedtotracksmanagedbyPDA— basedupdatealgorithms,whichareoutlinedinSection2. ItisimportaIlttonotethat,unlil(ethetargetvisibnity trackinitiationmethod.theclutter—basedtrackinitiation methodsdonotinfluencethebehavioroftheunderlying trackupdatealgorithminanywaThisdecouplinghasa numberofdesirablee珏lects.whicharediscussedinSections 4and5. 1工ackinitiati0nnlters 1.1Targetvisibilityfilter Theau皇atictmckinitiationtechniquedescrjbedin [7,10,26]isbrieflypresentedhereasareferenceIPDAiIn— plementation.Duringagivenupdateofagiventrack,the priorprobabi1ityoftargetvisibnityp(j七一1),estimated No.3KENNEDYHughLachlan:Clutter— basedTestStatisticsforAutomaticTrackInitiation267 recursivelyfrompreviousscans,isusedtoweightthepeak— is—due—to-targeteventsofthecurrentscank.Theposterior valueofP(klk)isthenupdatedusingBayesrule.The valueofPuispropagatedforwardintime,fromonescanto thenext,usingaMarkovchain, P(lk—1)=Ap(一llk—1)+t(1一P(一llk—1)) (1) withtheelements?and?c0rresp0nding(respectively) totheprobabilitythatthetargetremains,andbecomes, visible.TheMarkovchaineffectivelypreventsPfrombe— cominglockedatzeroorunityduringtherecursion,thus ensuringthatitremainsresponsivetonewdata.Trackcon— fidencepisusedforalltrackconfirmationanddeletion decisions;itisasmoothedrepresentati0nofPu, P(k)一(1一)P(klk)+P(k—1)(2) wherecisanarbitrarilyorempiricallychosensmooth— ingparameter,withnostatisticalorprobabilisticbasis.A tentativetrackisconfirmedwhenPc?Pc.n. 1.2Clutterdensityfilter Itisassumedthatclutterdetectionsareproducedby aPoissonprocess,sothattheprobabilitythatMclutter peaksarecontainedwithinagivenvolumeVisgivenby p.(M;p,):—(pV— )Mexp(-pV)(3) wheretheparameterPistheunitoccurrencerate.Dis— tancesbetweenconsecutivePoissoneventsaredistributed asanexponentialvariable.Thisrelationshipisusedto solvequeuingproblemsarisingininformationsystemst1. Whenappliedtotheproblemathandandusedtomodel theclutter—freevolume,withaclutterdensityofP,the exponentialdistributiontakesthefollowingform: ,D(;P)=pexp(一p)(4) Here,isthe(ellipsoida1)clutter—freespacearoundthe predictedmeasurementlocation,enclosedbythenearest associatedpeak.Itiscomputedusing = D,//lsl(5) whereSistheinnovationcovariancematrix;isthevol— umeoftheunitsphereinthem—dimensionalmeasurement space,computedusing andDistheMahalanobisdistance,computedusing D=, (6) (7) Intheaboveequation,istheinnovationvector,oflength mwiththesuperscripttdenotingitstranspose.Itisthe difrerencebetweenthepredictedvalueofthetargetmea- surementandtheobservedvalueofthenearesttargetmea- surement.ItisassumedthattheparameterPisknownbut notnecessarilyconstantfromscantoscan.Inpractice,it isusuallyestimatedbydividingthenumberofpeaksgen— eratedinagivenscan,whichisassumedheretobe large,bythesurveillancevolumeofthesensorfseel28I foranalternativeapproach1.AsaresuItoftherelationship betweenthegammaandchi—squareddistributions[ ,(4) yieldsthefollowingteststatistic: 2pV.q_x(2)(8) Ifatentativetrackisinitiatedatk=0.andupdatedusing thenearestdetectionoverthenextKconsecutivescans, thenfromthereproductivepropertyofchi?-squaredvari—? ables,thefollowingrelationshipholdsifthetrackisonly updatedbyclutterpeaks: K ZD()=2?p()()_x(2)(9)=1 wherethekindicesdenotethescandependenceofpand . Thenullhypothesis(H0D)isrejectedifZD(k)exceeds aspecifiedconfirmationthresholdD.whereDisselected togivethedesiredsizeOtD,usingtheinversechi—squared cumulativedensityfunctionfCDF).Thep-valueofthetest isusedasameasureoftrackconfidenceonthek—thscan. p.():厂.''_x(z;)dz(1o)J0 wherevisthedegreeoffreedom 1.3Clutteramplitudefilter Theamplitude(indB),orsignal—to—noiseratio(SNR), ofclutterpeaksabovethedetectionthresholdisassumed t0bedistributedasanexponentialvariablefforempirical suitabilityandanalyticalconvenience), ,A(a;OA,amin)=1exp{一a-amin)) wherefaistheprobabilitydensityfunction(PDF)ofthe distributi0n;amiistheSNRofthedetectionthreshold, asappliedbythepeakdetector;OAistheaverageSNR oftheportionoftheclutterpopulationthatisabovethe detectionthresholdandaistheSNRoftheclutterpeak 【aminSaSooJ. C0nsequently,thefollowingstatisticisdistributedasa chi—squaredvariable,with2degreesoffreedom: 2二 0A一0min_x(2)(12) Ifatentativetrackisinitiatedatk=0.thenreceivesK updates,andallassociatedpeaksareduetoclutter,then fromthereproductivepropertyofchi—squaredvariables,the followingstatisticresultsareobtained: 2 ?a(k,几) k:1 OA一0min_x(2K)(13) wherekisthescanindexand几istheindexofthenearest associatedpeakduringthekthscan.Similarly,iftheSNR 0fallotherpeaksfalsoassumedtobeduetoclutter1in eachscanaresummedthen 2 ? :ln amin~ OA一0min_x(2K(—1))(14) where?isthenumberofdetectionsreceivedineachscan. Theaminparameterisasystemconfigurationparameter itsvalueisthereforeknown;theOAparameterisunknown 0 rL , k ? ACTAAUT0MATICASINICAV_0l_34 andisdifficulttoestimatewhen?8issmall,however,itis assumedtobeconstantovertheKscans.Itiseliminated when(13)isdividedby(14)andateststatisticthatis distributedasSnedecor8Fvariableresultswhenthetwo equationsaredividedbytheirrespectivedegreesoffree— dom.Thefollowingholdswhenthenullhypothesisistrue: ZA()=(N一1)n),} —————————一 , ??{o(,n)一arai) n=l,n?n with ZA()F(2K,2K(一1))(15) Thenullhypothesis(H0^)isrejectedifZA(k)exceedsa specifiedconfirmationthresholdAA,whereAAisselected togivethedesiredsizeOtA,usingtheinverseFcumulative CDF.Thep-valueofthetestisused58ameasureoftrack confidenceonthekthscan, p()= . za(k) F(Z;Vl~v2)dz whereVlandv2arethedegreesoffreedom. 1.4Clutterfilterselectionandapplication (16) TosimplifythenotationinthederivationofZA,ithas beenassumedthatthenumberofdetectionsineach8ca4-1 isconstant,althoughthisneednotbethecase.Inboth clutter-basedmethods.afterthecreationofatentative track,theanalysiswindowlengthisallowedtogrowupto amaximumlet【gthofKmax;thereafter,aslidingwindowis used. Theclutterdensityfilteronlyyieldsalowandfalsetrack productionratewhenthespatialclutterdistributionisspa- tiallyuniformanduncorrelatedfrom8cantoscan.Anac— ceptableprobabilityoftruetrackpromotionisonlyob— tainediftheexpecteddistancetothetargetpeak,fromthe predictedpeaklocation,oneachupdate,onaverage,isless thantheexpecteddistancetothenearestclutterpeak. n1eclutteramplitudefilteronlyyieldsalowfalsetrack productionrateiftheincidenceofhigh—amplitudeclutter peaksisspatilyandtemporallyuncorrelated.Asatisfac— toryprobabilityoftruetrackpromotionisonlyrealizedif theaverageamplitudeoftargetpeaksisgreaterthanthe averageamplitudeofclutterpeaks. Bothclutterfiltersma?beappliedsimultaneouslyand independentlywheretheaforementionedpeakcondition8 prevail.Atentativetrackispromotedifbothoreithertest isre]ected.Eitherclutterfiltercanbeusedwithanyofthe trackupdatefiltersdescribedinthenextsection,58the trackupdateandtrackconfidencemethodsareindependent ofeachother. 2Trackupdatefilters 2.1PDA Theun-normalized(parametric)PDAeventprobabilities areevaluatedusing = (1一PdPg)(17) and =PdgD(zi;雪,S)(18) where雪isthepredictedtargetpeaklocation,Sisthein— novationcovariancematrix,ziisthelocationoftheith associatedpeak,andgDisthetargetlikelihoodfunction(a multivariateGaussian)inspatialcoordinates[引.Thenor— malizedPDAeventprobabilitiesarethenevaluatedusing = ,0,…,(19) ?, iI=0 Thesubscriptiisusedtoindextheeventprobabilities.with positiveindicesreferringtothei-thassociatedpeakand i=0correspondingtotheall—associated-peaks-are-clutter event,whichisassumedeithertobeduetoanabsenttarget detectionoratargetpeakthatisdetectedbutisbeyond thelimitsoftheassociationgate.TheparametersPdand Pgcorrespondtotheprobabilitythatthetargetpeaklies abovethedetectionthreshold,andtheprobabilitythatthe peaklieswithintheassociationgate,respectively.Here, PdisaknownconstantandPgiscomputedusingthelimit oftheellipsoidalassociationgate.F0llowingtheapproach takenin[30f,the?anearestpeaksarealwaysusedfi.e.a variablegatesize). Thean—normalizedeventprobabilitiesareweightedby afactor.,D(?c)whichisformedfromtheproductofuni- formprobabilitydensityfunctionsandaPoissonprobabil— itymassfunction,i.e., fD(No)=()×(唧c) TheargumentJ7vcisthenumberofclutterpeaksassumed intheeventhypothesis,i.e.,J7vc=?a一1,forthei>0 hypotheses,whereoneofthepeaksisassumedtobedue toatarget,andNc=N9forthei=0hypothesis,where allpeaksareassumedtobeduetoclutter.Afterdividing allhypothesesbylDLNonlyafactorofp}Ngremains inthei=0hypothesisin(18). 2.2PDAwithtargetvisibility As【101,thePDAeventspaceisextendedtoincludetrack initiationandterminationasfollows: and then fl"l=(1一P)P =P(1一PdPg)P (21) (22) =PvPdgD(z~;fl,S)(23) = ,i=--1,---,(24) ?, Intheaboveexpressions,P"representsthepriorproba- bilityoftargetvisibility(frompreviousreeursions),i.e., P(I一1);theposteriorprobabilityoftargetvisibilityis thenevaluatedusingP(I)=1一,1.PDAwithtarget visibilityisequivalenttoPDAwhenP"isfixedatunity. 2.3PDAwithtargetvisibilityandamplitudein_ formation Previousstudieshaveshownthattheconsiderationof amplitudeinformation,SNRorsignalstrength,inatrack- ingalgorithmhasthepotentialtojmproveoverlperfor- maglceofPDA—basedtrackingfilterst9,3l一341. Theextentof thebenefitsofcoursedependsontheexpectedexcessSNR ofthetargetpeaksrelativetotheclutterpeaks.Based No.3KENNEDYHughLachlan:Clutter— basedTestStatisticsforAutomaticTrackInitiation269 looselyontheapproachtakenin[30],SNRisincorporated3Simulations intothePDA--with--target--visibilityframeworkusing and fl"1:(1 =P(1 ,0i)(25) A,0m.)(26) Ng =Pdgo(zt;iJ,s)gA(.;)?fA(aj;OA,.…) J=1,j#i then : ?, i=一1 i:一1,?一, (27) (28) oruponrearranginganddividingalleventsbytheproduct ofa1lfAfunctions: and then : : Pd ()?( = {3 ?. E i=一l (29) (30) (31) i:一1,?一,(32) wherefAistheclutterlikelihoodfunctionfa?