Posterior Cramer-Rao Bounds for Multi-Target Tracking

Poseror Cramer-Rao Bounds for Mul-Targe Trackng C. HUE INRA France J-P. LE CADRE, Member, IEEE IRISA/CNRS France P. PÉREZ IRISA/INRIA France ACRONYMS B1 PCRB compued under he assumpon ha he assocaons are known B PCRB compued under he A1 and A assumpons B3 PCRB compued under he A1 and A3 assumpons CRB Cramér-Rao bounds PCRB Poseror Cramér-Rao bounds IRF Informaon reducon facor EM Expecaon-maxmzaon algorhm EKF Exended Kalman fler KF Kalman fler PDAF Probablsc daa assocaon fler JPDAF Jon probabls daa assocaon fler MHT Mulple hypoheses racker PMHT Probablsc mulple hypoheses racker MOPF Mulple objecs parcle fler RMSE Roo mean square error. Ths sudy s concerned wh mul-arge rackng (MTT. The Cramér-Rao lower bound (CRB s he basc ool for nvesgang esmaon performance. Though bascally defned for esmaon of deermnsc parameers, has been exended o sochasc ones n a Bayesan seng. In he arge rackng area, we have hus o deal wh he esmaon of he whole rajecory, self descrbed by a Markovan model. Ths leads up o he recursve formulaon of he poseror CRB (PCRB. The am of he work presened here s o exend hs calculaon of he PCRB o MTT under varous assumpons. NOTATIONS A º B A B posve sem-defne r X [(@=@ x1,:::,(@=@ xnx ] T Y T X r X r Y E p Expecaon compued w.r.. he densy p J (p E[ log(p] Leer used as an ndex o denoe me varyngbeween0andt Leer used as an exponen o denoe one of he M arges j Leer used as an exponen o denoe one of he m measuremens a me P d Deecon probably Parameer of he Posson law modelng he number of false alarms V observaon volume. I. INTRODUCTION Manuscrp receved June 4, 003; revsed February 14, 004 and Aprl 19, 005; released for publcaon May 1, 005. IEEE Log No. T-AES/4/1/870590. Refereeng of hs conrbuon was handled by P. K. Wlle. Auhors addresses: C. Hue, INRA, Cenre de Recherches de Toulouse, BP 7, F-3136, Casane, Tolosan Cedex, France, E-mal: (chue@toulouse.nra.fr; J-P. Le Cadre and P. Pérez, IRISA, Campus de Beauleu, 3504 Rennes Cedex, France. 0018-951/06/$17.00 c 006 IEEE Ths sudy s concerned wh mul-arge rackng (MTT,.e., he esmaon of he sae vecor made by concaenang he sae vecors of several arges. As assocaon beween measuremens and arges are unknown, MTT s much more complex han sngle-arge rackng. Exsng MTT algorhms generally presen wo basc ngredens: an esmaon algorhm coupled wh a daa assocaon mehod. Among he mos popular algorhms based on (exended Kalman flers (EKFs are he jon probablsc daa assocaon fler (JPDAF, he mulple hypohess racker (MHT or, more recenly, he probablsc MHT (PMHT. They vary on he assocaon mehod n use. Wh he developmen of he sequenal Mone Carlo (SMC mehods, new opporunes for MTT have appeared. The sae IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 4, NO. 1 JANUARY 006 37

dsrbuon s hen esmaed wh a fne weghed sum of Drac mass cenered around parcles. The Cramér-Rao lower bound (CRB [1] s wdely used for assessng esmaon performance. Though a grea deal of aenon has been pad o measures of performance such as rack 1 pury and correc assgnmen rao [] hese mehods are based on dscree assgnmens of measuremens o racks and are hus no unversally applcable. Ther neres s, o a large exen, due o he fac ha numerous MTT algorhms rely on hard assocaon. Whn hs framework hs ype of analyss s que pernen; bu here s a need for a smple and versale formulaon of a performance measure n he MTT conex; whch leads us o focus on CRB. These bounds are developed here n a general framework whch employs a probablsc srucure on he measuremen-o-arge assocaon. Agan, he dffculy of obanng CRB for MTT s due o a need for an assocaon beween measuremens and racks, and o ncorporae hs basc sep n he CRB calculaon. Thus, esmaon of he arge saes on he one hand, and of he measuremen-o-rack assocaon probables on he oher, are ghly relaed. On anoher hand, whle he CRB s an essenal ool for analyzng performance of deermnsc sysems, he poseror CRB (PCRB s a measure of he maxmum nformaon whch can be exraced from a dynamc sysem when boh measuremens and sae are assumed o be random, hus evaluang performance of he bes unbased fler. Thus, performance analyss s now consdered n a Bayesan seup. Naurally, hs analyss deals wh racks and dmenson grows lnearly wh me. Que remarkably, has been shown ha a recursve Rcca-lke formulaon of he PCRB could be derved under reasonable assumpons. Here, we show ha hs framework s sll vald n he MTT seup and allows us o derve convenen bounds. Ths paper s organzed as follows. The MTT problem s nroduced n Secon II, followed by a bref background on PCRB for nonlnear flerng (Secon III. Secon IV s he core of hs manuscrp snce deals wh he dervaon of he PCRB for MTT, under varous assocaon modelngs. These bounds are llusraed by compuaonal resuls. II. THE MULTI-TARGET TRACKING PROBLEM A. General Framework Le M be he number of arges o rack, assumed o be known and fxed here. The ndex desgnaes one among he M arges and s always used as 1 By rack, we consder here a sequence of saes assocaed wh a Markovan model. superscrp. MTT consss n esmang he sae vecor made by concaenang he sae vecors of all arges. I s generally assumed ha he arges are movng accordng o ndependen Markovan dynamcs, even hough can be crczed lke n [3]. A me, X =(X 1,:::,XM follows he sae equaon decomposed n M paral equaons: X = F (X 1,V 8 =1,:::,M: (1 The noses (V and(v0 are supposed only o be whe boh emporally and spaally, and ndependen for 6= 0. The observaon vecor colleced a me s denoed by y =(y 1,:::,ym. The ndex j s used as frs superscrp o refer o one of he m measuremens. The vecor y s composed of deecon measuremens and cluer measuremens. The false alarms are assumed o be unformly dsrbued n he observaon area. Ther number s assumed o arse from a Posson densy ¹ f of parameer V where V s he volume of he observaon area and he average number of false alarms per un volume. As we do no know he orgn of each measuremen, one has o nroduce he vecor K o descrbe he assocaons beween he measuremens and he arges. Each componen K j s a random varable ha akes s values among f0,:::,mg. Thus, K j = ndcaes ha y j s assocaed wh he h arge f =1,:::,M and ha s a false alarm f.inhe frs case, y j s a realzaon of he sochasc process: Y j = H (X,Wj f K j = : ( Agan, he noses (W j and(w j0 are supposed only o be whe noses, ndependen for j 6= j 0.Wedo no assocae any knemac model o false alarms. A measuremen recepon, he ndexng of he measuremens s arbrary and all he measuremens have he same pror probably o be assocaed wh a gven model. Thevarables(K j,:::,m are hen supposed dencally dsrbued. Ther common law s defned wh he probably (¼ =1,:::,M : ¼ =P(K j = 8 j =1,:::,m : (3 The probably ¼ s hen he pror probably ha an arbrary measuremen s assocaed wh model. The erm model denoes he arge f =1,:::,M and he model of false alarms f = 0. Inuvely, hs probably represens he observably of arge for =1,:::,M. The¼ vecor s consdered as a realzaon of he sochasc vecor =( 0, 1,:::, M wh he followng pror dsrbuon on : p( =p( 0 p( 1,:::, M j 0 (4 where p( 1,:::, M j 0 s unform on he hyperplane defned by P M =1 =1 0. 38 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 4, NO. 1 JANUARY 006

To solve he daa assocaon some assumpons are commonly made [4]: A1. One measuremen can orgnae from one arge or from he cluer. A. One arge can produce zero or one measuremen a one me. A3. One arge can produce zero or several measuremens a one me. Assumpon A1 expresses ha he assocaon s exclusve and exhausve. Unresolved observaons are hen excluded. From a mahemacal pon of vew, P he oal probably heorem can be used and M ¼ = 1 for every. Assumpon A mples for j =1,:::,m are dependen. Assumpon A3 s ofen crczed because may no mach he physcal realy. However, allows o suppose he sochasc ndependence of he varables ha he assocaon varables K j and drascally reduces he complexy of he ¼ vecor esmaon. K j B. Revew of Man MTT Algorhms Le us now brefly revew he reamen of he daa assocaon problem. The followng algorhms essenally dffer accordng o her esmaon srucure (deermnsc or sochasc and her assocaon assumpons. Frs, he daa assocaon problem occurs as soon as here s uncerany n measuremen orgn and no only n he case of mulple arges. In he case of one sngle-arge rackng, he negraon of false alarms n he model hen mples daa assocaon. The probablsc daa assocaon fler (PDAF [5] akes no accoun hs uncerany under he classcal hypoheses A1 and A. The JPDAF s an exenson of he PDAF for mulple arges [6]. Boh hese algorhms are based on Kalman fler (KF and consequenly assume lnear models and addve Gaussan noses n (1 and (. The man approxmaon consss of assumng ha he predced law s sll Gaussan whereas s n realy a sum of Gaussan assocaed wh he dfferen assocaons. The MHT sll uses A1 and A bu allows he deecon of a new arge a each me sep [7]. To cope wh he exploson of he assocaon number, some of hem mus be gnored n he esmaon. For hese hree algorhms ((JPDAF, MHT, a pror sascal valdaon of he measuremens decreases he nal assocaon number. Ths valdaon s based on he fundamenal hypohess ha he law p(y j Y 1: 1 s Gaussan, cenered around he predced measuremen and wh he nnovaon covarance. The valdaon gae s hen usually defned as he measuremen se for whch he Mahalanobs dsance o he predced measuremen s lower han a ceran hreshold. Some deals can be found n [4] TABLE I Classfcaon of Man MTT Algorhms Accordng o Ther Assocaon Assumpon and Esmaon Srucure Assocaon Assumpon Esmaon srucure A1 A A1 A3 Kalman fler (JPDAF MHT EM PMHT parcle fler SIR-JPDAF MOPF for nsance. Ths valdaon gae procedure wll no be consdered hroughou, whch means ha all he measuremens wll be aken no accoun. Unlke he above algorhms, he PMHT s based on he assumpons A1 and A3. I proposes he bach esmaon of mulple arges n cluer va an expecaon-maxmzaon (EM algorhm. Radcally dfferen from a deermnsc approach lke KF-based rackers or EM-based rackers, he sochasc approach developed quckly hese las years. SMC mehods [8] esmae he enre a poseror law of he saes and no only he frs momens of hs law lke KF-based rackers do. In he conex of MTT, parcle flers are parcularly appealng: as he assocaon needs only o be consdered a a gven me eraon, he complexy of daa assocaon s reduced. For a sae of ar of he proposed algorhms he reader can refer o [9]. Agan, we can dsngush algorhms usng A for solvng daa assocaon lke he sequenal mporance resamplng (JPDAF, SIR-JPDAF [10] or usng A3 lke he mulple objecs parcle fler (MOPF [11]. Classfcaon of he above algorhms accordng o her assocaon assumpon and esmaon srucure are summarzed n Table I. III. BACKGROUND ON POSTERIOR CRAMÉR-RAO BOUNDS FOR NONLINEAR FILTERING I s of grea neres o derve mnmum varance bounds on esmaon errors o have an dea of he maxmum knowledge on he saes ha can be expeced and o assess he qualy of he resuls of he proposed algorhms compared wh he bounds. Frs defned and used n he conex of consan parameer esmaon, he nverse of he Fsher nformaon marx, commonly called he Cramér-Rao (CR bound, has been exended o he case of random parameer esmaon n [1], hen called he PCRB. Le X R n x be a sochasc vecor and Y R n y asochasc observaon vecor. The mean-square error of any esmae ˆX(Y sasfes he nequaly E( ˆX(Y X( ˆX(Y X T º J 1 (5 The nequaly means ha he dfference E( ˆX(Y X( ˆX(Y X T J 1 s a posve sem-defne marx. HUE ET AL.: POSTERIOR CRAMER-RAO BOUNDS FOR MULTI-TARGET TRACKING 39

where J = E @ logp X,Y (X,Y=@X s he Fsher nformaon marx and where he expecaons are w.r.. he jon densy p X,Y (X,Y under he followng condons. 1 @p X,Y (X,Y=@X and @ p X,Y (X,Y=@X exs and are absoluely negrable w.r.. X and Y. The esmaor bas Z B(X= ( ˆX(Y Xp YjX (Y j XdY R ny sasfes: lm B(Xp(X, 8 l =1,:::,n x : X l! 1 (6 Le us consder he nonlnear dscree sysem for a unque objec: ½ X = F (X 1,V (7 Y = H (X,W and he assocaed flerng problem,.e., he esmaon of X gven Y 0: =(Y0,:::,Y. A frs approach consss of usng a lnear Gaussan sysem equvalen o (7 lke n [1] and [13]. The error covarance of he nal sysem s hen lower bounded by he error covarance of he Gaussan sysem. Neverheless, wo major remarks can be made [14]. Frs, he equvalen noon s no precsely defned n [1] and [13]. Second, seems no lkely ha here always exss such a lnear Gaussan sysem for nsance f he probably densy funcon (pdf s mulmodal. A revew of hs approach can be found n [14]. The approach recenly developed by Tchavsky, e al. n [15] orgnally consders he Fsher nformaon marx for he esmaon of X gven Y 0: as a submarx of he Fsher nformaon marx assocaed wh he esmaon of X 0: gven Y 0:. Usng he noaons of [15], J(X 0: denoes he (( +1n x ( +1n x nformaon marx of X 0: and J X denoes he n x n x nformaon submarx of X whch s he nverse of he n x n x rgh lower block of [J(X 0: ] 1. To avod nverson of oo large marces, a recursve expresson of he bound J X has been presenedrecenlyn[15]and[16]andsummarzed by he followng formula: J X+1 = DX DX 1 (J X + DX 11 1 DX 1 (8 where DX 11 = E[ X X logp(x +1 j X ] D 1 X = E[ X +1 X logp(x +1 j X ] DX 1 = E[ X X +1 logp(x +1 j X ] = [DX 1 ] T D X = E[ X +1 X +1 logp(x +1 j X ] + E[ X +1 X +1 logp(y +1 j X +1 ] (9 and where he r and operaors denoe he frs and second paral dervaves, respecvely: # T @ @ r X =,:::,, Y X @ = r X r Y T : (10 x1 @ xnx The marx J 1 X +1 provdes a lower bound on he mean-square error of esmang X +1. I can be shown n [17] ha hs bound s overopmsc bu has he grea advanage o be recursvely compuable. Le us see now some exensons recenly proposed for he PCRB. A. Inegraon of Deecon Probably In [18], he auhors propose o negrae he deecon probably n he prevous bound. For a scenaro of gven lengh, he bound s compued as a weghed sum on every possble deecon/nondeecon sequence. As he number of erms of hs sum grows exponenally he less sgnfcan are no aken no accoun. B. Exenson o Measuremen Orgn Unceranes Several works have suded CRBs for models wh measuremen orgn unceranes, bu for a sngle-arge. The assocaon of each measuremen o he arge or o he false alarm model can be done under he classcal hypoheses A1 and A or under A1 and A3. As CRB was frs defned for parameer esmaon, models wh deermnsc rajecores have frs been suded. If he nose s Gaussan, has been shown n [19] and [0] ha, under A1 and A, he nverse of he nformaon marx can hen be wren as he produc of he nverse of he nformaon marx whou false alarms by an nformaon reducon facor, noed IRF and lower han uny. In [1], he auhors show ha here s also an IRF for he PMHT measuremen model,.e., under he hypoheses A1 and A3. In he case of dynamc models, he exenson of he bound (8 o he case of lnear and nonlnear flerng wh measuremen orgn uncerany due o cluer has been recenly suded n [] and [3]. The exenson manly consss of replacng he classc pdf of he measuremen gven he sae by he pdf of he measuremen vecor akng no accoun he measuremen uncerany. The conclusons are he followng. 1 Under he assumpon of a Gaussan observaon nose wh a dagonal covarance marx, an IRF dagonal marx appears n he PCRB. The PCRB does no show nsably whereas rackng algorhms can relavely easly be pu no wrong. 40 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 4, NO. 1 JANUARY 006

3 The PCRB would be more affeced by a low P d han by a hgh sae or nose covarance or by a hgh cluer densy. 4 For low deecon probables, he PCRB s really overopmsc (versus PDAF RMSE. IV. POSTERIOR CRAMÉR-RAO BOUNDS FOR MULTI-TARGET TRACKING Now, le us see how he PCRB proposed n [15] can be exended and used n he case of mulple arges flerng defned by (1 and (. Noe ha n hs case, he measuremen vecor s composed of deecon measuremens ssued from he dfferen arges and of false alarms. The followng exenson hen akes no accoun smulaneously he measuremen uncerany and he exenson of one o mulple arges. Frs, he recursve equaon (8 can be obaned as well for mulple arges usng he srucure of he jon law: p(x 0:+1,Y 0:+1 =p(x 0:,Y 0: p(x +1 j X p(y +1 j X +1 : Ths srucure s sll rue for mulple arges, whch leads o he same recursve formula for he nformaon marx. As he arges are supposed o move accordng o ndependen dynamcs, we have logp(x 1:M +1 j X1:M = MX =1 (11 logp(x+1 j X : (1 Consequenly, he marces DX 11, DX 1 and he frs erm of DX are smply block-dagonal marces where he h block s compued w.r.. X and X+1.Iremanshe second erm of DX,.e.,E[ X1:M +1 logp(y X 1:M +1 j X+1 1:M]. +1 As n [], we can decompose hs erm accordng o he observaon number usng he oal probably heorem: E[ X1:M +1 logp(y X 1:M +1 j X+1 1:M ] +1 1X = P(m +1 E[ X1:M +1 logp(y m +1 X 1:M +1 j X+1 1:M ] : +1 m +1 =1 B(m +1 The probables P(m +1 aregvenby P(m +1 = ¹= d (13 ¹X ( V d exp V P ¹ d d : (14 d! To compue B(m +1, we have o face agan he assocaon problem: some addonnal hypoheses mus be formulaed o gve explc expressons of he lkelhood p(y m +1 +1 j X +1. The problem s ha hese hypoheses condon he esmaon algorhm, whle hey should no nfluence he heorecal bound. We propose here o derve hree bounds: B1, he PCRB compued under he assumpon ha he assocaons are known. B, he PCRB compued under he A1 and A assumpons. B3, he PCRB compued under he A1 and A3 assumpons. The followng lemma s used hroughou he sequel. LEMMA 1 Le X =(X 1,:::,X M R n x and Y R n y wo sochasc varables and 1, wo negers [1,:::, M], hen he followng expecaon equaly holds rue: E X E YjX [ X logp(y j X] X 1 = E X E YjX [r logp(y j X(r X X logp(y j 1 XT ]: Le us defne he followng noaon: for wo vecors, and p a probably law, (15 J (p =E[ log(p]: (16 In he nex hree paragraphs we descrbe J X1:M +1 (p(y m X 1:M +1 j X1:M +1 accordng o he assocaon +1 assumpons. A. PCRB B1 The assocaon vecor s supposed o be known. We hen have Xm logp(y = y m j X = x,k = k = logp(y j j x kj : (17 The graden of he log-lkelhood w.r.. X s no zero only f here exss j such ha k j =. Inhs case, r X logp(y m j x,k = r X p(yj j x : (18 p(y j j x We fnally oban for all =1,:::,M: J X (p(y m X and r Xp(y j j x j x,k = E X E j (r X p(y j j x T Y jx p(y j j x (19 J X (p(y X 1 j x,k = 0 f 1 6= : (0 HUE ET AL.: POSTERIOR CRAMER-RAO BOUNDS FOR MULTI-TARGET TRACKING 41

B. PCRB B We can wre logp(y = y m j X = x A1 A = log X k p(y =(y 1,:::,ym j x,k p(k =log X k Ym p(y j j x,k p(k : (1 The probably p(k = k can be compued from he deecon probably P d, he number of false alarms ª k, her dsrbuon law ¹ f and he bnary varable D K ( equal o one f he objec s deeced, zero else: p(k = k = ª k! m! ¹ f (ª k MY =1 P DK ( d MY (1 P d (: D1 K ( The graden of he log-lkelhood w.r.. X s Q Pk r X logp(y j x = r m X p(yj j x,k p(k : p(y j x (3 Le us denoe by k ¾ he assocaons ha assocae one measuremen o he h arge. Under A, here exss a mos one such measuremen, denoed j. Then, P k ¾ Qj6=j p(y j r X logp(y j x = j x,k p(k r X p(y j j x : p(y j x (4 Usng Lemma 1, we oban for all 1, =1,:::,M: =1 PMHT, he maxmzaon sep for ¼ depends on he preceden esmaes for X and vce versa. The esmaon qualy of one hen srongly affecs he esmaon qualy of he oher. Smlarly for he MOPF, he smulaed values for ¼ are used for smulaed X values and vce versa. In hs conex, seems o us naural o consder he PCRB for he esmaon of he jon vecor (,X. For all ha, he PCRB on he esmaon of X can be deduced from he global one by an nverson formula as we see laer. From he equaly P M ¼ =1andas¼0 s fxed a each nsan, we only consder he M 1 componens 1:M 1 =( 1,:::, M 1. Le us defne =( 1:M 1 ; he jon law s,x 1:M p +1 =p( 0:+1,Y 0:+1 =p p(y +1 j +1 p(x +1 j X p( +1 : (6 Le J( 0: be he nformaon marx of 0: assocaed wh p ; we are neresed n a recursve expresson on of he nformaon submarx J for esmang. Le us recall ha J s he nformaon submarx of whch s he nverse of he rgh lower block of [J( 0: ] 1. Usng he srucure of he jon law p +1 and he same argumen as n [15], he followng recursve formula can be shown (see he proof n he appendx: J +1 = D D 1 (J + D 11 1 D 1 (7 where 0 D 11 = J 0 (p(x +1 j X = 0 DX 11 0 D 1 = J +1 0 (p(x +1 j X = 0 DX 1 E[ X logp(y X 1 j X ] 6 = E X E Y jx 4 Pk ¾ 1 Q j6=j 1 p(yj j x,k p(k r X p(y 1 j1 j x 1 p(y j x X Y k ¾ p(y j j x,k p(k (r X j6=j p(y j 3 7 j x T 5 (5 where E X and E Y jx denoe, respecvely, he expecaon w.r.. he densy p(x andp(y j X. Le us noce ha he negrals w.r.. y are m n y -dmensonal. C. PCRB B3 To our knowledge, algorhms usng A3 need a jon esmaon of X and ¼.Inhsway,forhe D = J +1 +1 (p(y +1 j +1 p(x +1 j X p( +1 (8 0 0 = 0 J X +1 X +1 (p(x +1 j X J +1 + +1 (p( +1 0 0 0 + J +1 +1 (p(y +1 j +1 : Once J s recursvely compued, a lower bound on he mean-square error of esmang X sgvenbyhe 4 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 4, NO. 1 JANUARY 006

nverson formula appled o he rgh lower block J X of J J X # J = : J X E( ˆX(Y X( ˆX(Y X T º [J X J X J 1 J X ] 1 : (9 As a unform pror s assumed for he law, J +1 +1 (p( +1 s zero. To evaluae he hrd erm of D,wecanwre J X and he same expressons for 1 = M by replacng ¼ M by 1 P M 1 ¼. Noce ha under hese assocaon assumpons, all he negrals w.r.. y j are n y -dmensonal. D. Mone Carlo Evaluaon for a Bearngs-Only Applcaon Leusbegnwhhecasewhereheevoluon model s lnear and Gaussan. As n [15], we logp(y = y j = Á A1 A3 Ym = log p(y j j Á # Xm ¼ 0 M 1 = log V X ¼0 p(yj j x M + (p(y j j x p(yj j x M ¼ + p(yj j x M : (30 =1 For 6= M, he graden w.r.. X s Xm r r X logp(y j Á =¼ X p(y j j x p(y j : (31 j Á A smlar expresson for = M s obaned by replacng ¼ M by 1 P M 1 ¼.For =1,:::,M 1: Xm p(y j j x r logp(y j Á = p(yj j x M p(y j : (3 j Á Usng Lemma 1, we oban for 1, 6= M J X (p(y X 1 j =E[r X 1 (r X = E 4 ¼ 1 ¼ logp(y j T ] m X r p(y j j x 1 E X1 j Y j (r X p(y j j x p(y j j Á T 3 5 analycally oban he followng equales: DX 11 = dagff T V 1 F g, 3 DX 1 =dagf F T V 1 g and J X +1 X +1 (p(x +1 j X = dagf V 1 g. In he general case of an observaon model wh an addve Gaussan nose defned as follows: p(y j j x we have =(¼ ny de 1= exp f 1 (yj H(x T 1 (y j H(x g r X p(y 1 j j x 1 =p(yj j x 1 I reads for he PCRB B1: r X 1 (36 H T (x 1 1 (y j H(x 1 : J X (p(y X j X = E X r X H T (x 1 (r X H T (x T (37 (38 (33 and he same expressons for 1 or = M by replacng ¼ M by 1 P M 1 ¼. For 1, 6= M: 3.e., he block-dagonal marx whose h block s equal o F T 1 V F. J 1 Xm (p(y j = E 4 E Y j j (p(y j j x 1 p(yj j x M # 3 (p(y j j x p(yj j x M 5: p(y j (34 j Á For 1, 6= M: J (p(y X 1 j = E 4¼ 1 Xm E Y j j p(y j j x p(yj j x M p(y j r j Á X 1 p(y j j x 1 #3 5 (35 HUE ET AL.: POSTERIOR CRAMER-RAO BOUNDS FOR MULTI-TARGET TRACKING 43

for he PCRB B: J X X 1 P (p(y j X = E X r X H T (x 1 1 1 k E ¾ p(y 1 j x,k p(k (yj1 H(x 1 Y jx p(y j x X k ¾ p(y j x,k p(k (y j H(x # T 1 (r X H T (x T # (39 and for he PCRB B3: J X (p(y X 1 j = E ¼ 1 ¼ J (p(y X 1 j = E 4¼ 1 r X 1 H T (x 1 1 Xm E Y j j p(y j j x 1 p(yj j x p(y j j Á (y j H(x 1 (yj H(x T # 1 (r X H T (x #: T (40 r X 1 H T (x 1 1 Xm E Y j j p(y j j x p(yj j x M p(y j j Á p(y j j x 1 (yj H(x 1 #3 5: (41 In he bearngs-only applcaon, we have n y = 1andhenH T = H ha leads o some wrng smplfcaons. We deal wh classcal bearngs-only expermens wh hree arges. In he conex of a slowly maneuverng arge, we have chosen a nearly-consan-velocy model. 1 The Scenaro: The sae vecor X represens he coordnaes and he veloces n he x-y plane: X =(x,y,vx,vy for = 1,,3. For each arge, he dscrezed sae equaon assocaed wh me perod s µ X+ = I I X 0 I + 0 @ I 0 1 AV 0 I (4 where I s he deny marx n dmenson and V s a Gaussan zero-mean vecor wh covarance marx V =dag[¾x,¾ y,¾ x,¾ y ]. A se of m measuremens s avalable a dscree mes and can be dvded no wo subses. 1 One subse s of rue measuremens whch follow (43. A measuremen produced by he h arge s generaed accordng o Y j =arcan µ y y obs x + W j x obs (43 where W j s a zero-mean Gaussan nose wh covarance ¾w :05 rad ndependen of V,andx obs and y obs are he Caresan coordnaes of he observer, whch are known. We assume ha he measuremen produced by one arge s avalable wh a deecon probably P d. The oher subse s of false measuremens whose number follows a Posson dsrbuon wh mean V where s he mean number of false alarms per un volume. We assume hese false alarms are ndependen and unformly dsrbued whn he observaon volume V. The nal coordnaes of he arges and of he observer are he followng (n meer and meer/second, respecvely: X 1 0 = (00,1500,1, 0:5T, X 3 0 =( 00, 1500,1,0:5T X 0 = (0,0,1,0T X0 obs = (00, 3000,1:,0:5 T : (44 The observer s followng a leg-by-leg rajecory. Is velocy vecor s consan on each leg and modfed a he followng nsans, so ha: Ã! vx obs µ 00,600,900 0:6 = vy00,600,900 obs à vx obs 400,800 vy obs 400,800 0:3! µ :0 = 0:3 : The rajecores of he hree objecs and of he observer are ploed n Fg. 1(a. E. The Assocaed PCRB The hree bounds are frs nalzed o J X0 = P 1 X 0 for B1 andb andj 0 = P 1 0 for B3 wherep X0 = (45 44 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 4, NO. 1 JANUARY 006

Fg. 1. (a Trajecores of he hree arges and of he observer. (b Measuremens smulaed wh P d :9 and V =3. dagfxcov g wh X cov =dagf150,150,0:1,0:1g and P 0 =dagfdagf0:05, =1,:::,M 1g;P X0 g. Then, o esmae he marces needed n he recurson formulas (8 or (7, we perform Mone Carlo negraon by carryng ou P1 ndependen sae rajecores and for each of hem P ndependen measuremen realzaons, and addonally P3 ndependen realzaons of he ¼ vecor for he PCRB B3 (P1, P, and P3 have been fxed o 100 n he followng compuaons. For nsance, he esmae Ĵ X of J X s compued as X 1 X 1 where J(x p1 Ĵ X X 1 = 1 P1P XP1 XP p1=1 p=1 J(x p1,y p1,p (46,y p1,p s he quany whose expecaon s o be compued n (39. We hen obaned he marx nequales: E( 1:M 1:M ˆX +1 (Y X( ˆX +1 (Y XT º B for =1,,3: (47 In he scenaro descrbed above, he marces B dmenson s equal o dm = 3 4 = 1. To nerpre he nequales (47, we have derved he scalar mean-square error gven by he race of (47: 1:M E( ˆX +1 (Y XT 1:M ( ˆX +1 (Y X rb (48 and he nequaly on he volume of he marces defned as he deermnan a he power 1=dm: [dee( 1:M 1:M ˆX +1 (Y X( ˆX +1 (Y XT ] 1=dm º [deb ] 1=dm : (49 We have compued he race and he volume of he hree bounds for dfferen values of he parameers ¾ x, ¾ y, P d, V. Frs, for a dynamc nose sandard ¾ x = ¾ y :0005 ms 1, a deecon probably P d :9 and V = 1,,3, he race and he volume are ploed agans me on he hree frs rows of Fg.. The resuls on he fourh row have been obaned for a hgher dynamc nose sandard ¾ x = ¾ y :001 ms 1, P d :9 and V =1.Theffh and las row corresponds o a scenaro where a deecon hole s smulaed for he frs objec durng a hundred consecuve nsans, beween mes 600 and 700. Whaever he parameers values, he nsan or he funcon f of he bounds consdered (race or volume, we always have f(b f(b3 f(b1 wh a greaer gap beween f(b3 and f(b1 han beween f(b and f(b3. More precsely, frs means ha he opmal performance whch can be obaned wh an algorhm usng assumpons A1 and A are below he opmal performance whch can be obaned wh an algorhm usng assumpons A1 and A3. Second, he opmal performance obaned wh an algorhm assumng he assocaon s known s far beer han for he wo precedng cases. For all ha, nohng can be concluded on he relave performance of he SIR-JPDA and of he MOPF for nsance. Such sudy needs he esmaon of he RMSE of boh algorhms over a hgh number of realzaons of he process and measuremen nose. For each couple of realzaon of boh noses, several runs of he algorhms are needed. To go back over he analyss of Fg., he plos presen wo peaks around mes 150 and 400. They correspond o nsans where bearngs from he hree arges are very close as shown n Fg. 1(b for one parcular realzaon of he rajecores and of he measuremens. Durng he second peak, he gap beween B and B3 on he one hand and B1 on he oher hand s wdenng. A slgh peak s also observed when he frs arge s no deeced (see las row of Fg.. Fnally, by comparng he hree frs rows, we observe ha he gap beween f(b and f(b3 s wdenng wh he cluer densy V. In all hese scenaros, as he deecon probably P d s srcly nferor o uny, may happen a one nsan ha no arge s deeced. If moreover no cluer measuremen s smulaed a ha nsan, he measuremen vecor Y s empy. In hs case, we smply se he expecaons J X +1 X +1 (p(y +1 j X +1 and J +1 +1 (p( +1 j X +1 o zero and he recursve formula (8 and (7 are reduced. V. CONCLUSION In hs manuscrp, an exenson of he PCRB from a sngle-arge o mul-arge flerng problem HUE ET AL.: POSTERIOR CRAMER-RAO BOUNDS FOR MULTI-TARGET TRACKING 45

Fg.. Trace and volume of he hree PCRB marces: B (dashed,b3 (sold,b1 (dashdoed. Lef column: race. Rgh column: volume. Frs (op row: ¾ x = ¾ y :0005 ms 1 and V = 1. Second row: ¾ x = ¾ y :0005 ms 1 and V =. Thrd row: ¾ x = ¾ y :0005 ms 1 and V = 3. Fourh row: ¾ x = ¾ y :0001 ms 1 and V = 1. Ffh (boom row: ¾ x = ¾ y :0005 ms 1 and a deecon hole beween mes 600 and 700 for objec 1. 46 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 4, NO. 1 JANUARY 006

has been suded. Three bounds have been derved accordng o he assocaon assumpons beween he measuremens and he arges. Based on Mone Carlo negraon, esmaes of hese hree bounds have fnally been proposed and evaluaed for he bearngs-only applcaon. APPENDIX. RECURSIVE FORMULA OF PCRB B By defnon, he nformaon marx J( 0:+1 of 0:+1 assocaed wh he law p +1 can be expressed as J( 0:+1 = 6 4 J 0: 1 0: 1 (p +1 J 0: 1 (p +1 J +1 3 0: 1 (p +1 7 5 J 0: 1 (p +1 J (p +1 J +1 (p +1 J 0: 1 +1 (p +1 J +1 (p +1 J +1 +1 (p +1 where J (p =E[ log(p]. Usng (6, reads J 0: 1 0: 1 (p +1 =J 0: 1 0: 1 (p J 0: 1 (p +1 =J 0: 1 (p J 0: 1 +1 (p +1 =J 0: 1 +1 (p (50 + J 0: 1 0: 1 (p(y +1 j +1 p(x +1 j X p( +1 (51 + J 0: 1 (p(y +1 j +1 p(x +1 j X p( +1 (5 + J 0: 1 (p(y +1 j +1 p(x +1 j X p( +1 J (p +1 =J (p +J (p(x +1 j X + J (p(y +1 j +1 p( +1 J +1 (p +1 =J +1 (p(x +1 j X + J +1 (p J +1 +1 (p +1 =J +1 +1 (p + J +1 (p(y +1 j +1 p( +1 + J +1 +1 (p(y +1 j +1 p(x +1 j X p( +1 : (53 (54 (55 (56 Usng (51 (56 and he noaon: A B J( 0: = we have he recursve formula: A B 0 6 J( 0:+1 = 4B C + D 11 where D 11 = J (p(x +1 j X D 1 = J +1 (p(x +1 j X B T C 0 D 1 T D 1 D D = J +1 +1 (p(y +1 j +1 p(x +1 j X p( +1 : (57 3 7 5 (58 (59 Now, J +1 s he nverse of he rgh lower block of J( 0:+1 1. Usng wce a classcal nverson lemma, we oban 1 A J +1 = D [0 D 1 B 0 ] REFERENCES B T C + D 11 = D D 1 [C + D 11 B T A 1 B ] 1 D 1 D 1 = D D 1 [J + D 11 ] 1 D 1 : (60 [1] Van Trees, H. L. Deecon, Esmaon, and Modulaon Theory (Par I. New York: Wley, 1968. [] Chang, K. C., Mor, S. and Chong, C. Y. Performance evaluaon of rack naon n dense arge envronmens. IEEE Transacons on Aerospace and Elecronc Sysems, 30, 1 (1994, 13 18. [3] Mahler, R. Mul-source mul-arge flerng: A unfed approach. SPIE Proceedngs, 3373 (1998, 96 307. [4] Bar-Shalom, Y., and Formann, T. E. Trackng and daa assocaon. New York: Academc Press, 1988. [5] Bar-Shalom, Y., and Tse, E. Trackng n a cluered envronmen wh probablsc daa assocaon. In Proceedngs of he 4h Symposum on Nonlnear Esmaon Theory and s Applcaons, 1973. [6] Formann, T. E., Bar-Shalom, Y., and Scheffe, M. Sonar rackng of mulple arges usng jon probablsc daa assocaon. IEEE Journal of Oceanc Engneerng, 8 (July 1983, 173 184. [7] Red, D. An algorhm for rackng mulple arges. IEEE Transacons on Auomaon and Conrol, 4, 6 (1979, 84 90. [8] Douce, A., De Freas, N., and Gordon, N. (Eds. Sequenal Mone Carlo Mehods n Pracce. New York: Sprnger, 001. [9] Hue, C., Le Cadre, J-P., and Pérez, P. Sequenal Mone Carlo mehods for mulple arge rackng and daa fuson. IEEE Transacons on Sgnal Processng, 50, (Feb. 00, 309 35. HUE ET AL.: POSTERIOR CRAMER-RAO BOUNDS FOR MULTI-TARGET TRACKING 47

[10] Oron, M., and Fzgerald, W. A Bayesan approach o rackng mulple arges usng sensor arrays and parcle flers. IEEE Transacons on Sgnal Processng, 50, (00, 16 3. [11] Hue, C., Le Cadre, J-P., and Pérez, P. Trackng mulple objecs wh parcle flerng. IEEE Transacons on Aerospace and Elecronc Sysems, 38, 3 (July 00, 791 81. [1] Bobrovsky, B. Z., and Zaka, M. A lower bound on he esmaon error for Markov processes. IEEE Transacons on Auomac Conrol, 0, 6 (Dec. 1975, 785 788. [13] Galdos, J. I. ACramér-Rao bound for muldmensonal dscree-me dynamcal sysems. IEEE Transacons on Auomac Conrol, 5, 1 (1980, 117 119. [14] Kerr, T. H. Saus of Cramér-Rao-lke lower bounds for nonlnear flerng. IEEE Transacons on Aerospace and Elecronc Sysems, 5, 5 (Sep. 1989, 590 600. [15] Tchavský, P., Muravchk, C., and Nehora, A. Poseror Cramér-Rao bounds for dscree-me nonlnear flerng. IEEE Transacons on Sgnal Processng, 46, 5(May 1998, 1386 1396. [16] Bergman, N. Recursve Bayesan esmaon: Navgaon and rackng applcaons. Ph.D. dsseraon, Lnköpng Unversy, Sweden, 1999. [17] Bobrovsky, B. Z., Mayer-Wolf, E., and Zaka, M. Some classes of global Cramér-Rao bounds. The Annals of Sascs, 15, 4 (1987, 141 1438. [18] Farna, A., Rsc, B., and Tmmoner, L. Cramér-Rao bound for non lnear flerng wh P d < 1 and s applcaon o arge rackng. IEEE Transacons on Sgnal Processng, 50, 8 (00, 1916 194. [19] Jauffre, C., and Bar-Shalom, Y. Track formaon wh bearng and frequency measuremens n cluer. IEEE Transacons on Aerospace and Elecroncs, 6, 6 (1990, 999 1009. [0] Krubajan, T., and Bar-Shalom, Y. Low observable arge moon analyss usng amplude nformaon. IEEE Transacons on Aerospace and Elecorncs, 3, 4 (1996, 1367 1384. [1] Ruan, Y., Wlle, P., and Sre, R. A comparson of he PMHT and PDAF rackng algorhms based on her model CRLBs. In Proceedngs of SPIE Aerosense Conference on Acquson, Trackng and Ponng, Orlando, FL, Apr. 1999. [] Zhang, X., and Wlle, P. Cramér-Rao bounds for dscree-me lnear flerng wh measuremen orgn unceranes. In Workshop on Esmaon, Trackng, and Fuson: A Trbue o Yaakov Bar-Shalom, May 001. [3] Hernandez, M., Marrs, A., Gordon, N., Maskell, S., and Reed, C. Cramér-Rao bounds for nonlnear flerng wh measuremen orgn uncerany. In Proceedngs of 5h Inernaonal Conference on Informaon Fuson, July 00. 48 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 4, NO. 1 JANUARY 006

Carne Hue was born n 1977. She receved he M.Sc. degree n mahemacs and compuer scence n 1999 and he Ph.D. degree n appled mahemacs n 003, boh from he Unversy of Rennes, France. Snce he end of 003 she has been a full-me researcher a INRA, he French Naonal Insue for Agrculural Research. Her research neress nclude sascal mehods for model calbraon, daa assmlaon, sensvy analyss, and n parcular, he Bayesan approach for agronomc models. Jean-Perre Le Cadre (M 93 receved he M.S. degree n mahemacs n 1977, he Docora de 3 eme cycle n 198, and he Docora d Ea n 1987, boh from INPG, Grenoble. From 1980 o 1989, he worked a he GERDSM (Groupe d Eudes e de Recherche en Deecon Sous-Marne, a laboraory of he DCN (Drecon des Consrucons Navales, manly on array processng. Snce 1989, he s wh IRISA/CNRS, where he s Dreceur de Recherche a CNRS. Hs neress are now opcs lke sysem analyss, deecon, mularge rackng, daa assocaon, and operaons research. Dr. Le Cadre has receved (wh O. Zugmeyer he Eurasp Sgnal Processng bes paper award (1993. Parck Pérez was born n 1968. He graduaed from ÉcoleCenralePars,France, n 1990 and receved he Ph.D. degree from he Unversy of Rennes, France, n 1993. Afer one year as an Inra pos-docoral researcher n he Deparmen of Appled Mahemacs a Brown Unversy, Provdence, RI, he was apponed a Inra n 1994 as a full me researcher. From 000 o 004, he was wh Mcrosof Research n Cambrdge, U.K. In 004, he became senor researcher a Inra, and he s now wh he Vsa research group a Irsa/Inra-Rennes. Hs research neress nclude probablsc models for undersandng, analysng, and manpulang sll and movng mages. HUE ET AL.: POSTERIOR CRAMER-RAO BOUNDS FOR MULTI-TARGET TRACKING 49