A Formulation of Multitarget Tracking as an Incomplete Data Problem

Transcription

1 I. INTRODUCTION A Forulation of Multitarget Tracing as an Incoplete Data Proble H. GAUVRIT J. P LE CADRE IRISA/CNRS France C. JAUFFRET DCN/Ingénierie/Sud France Traditional ultihypothesis tracing ethods rely upon an enueration of all the assignents of easureents to tracs. Pruning and gating are used to retain only the ost liely hypotheses in order to drastically liit the set of feasible associations. The ain ris is to eliinate correct easureent sequences. The Probabilistic Multiple Hypothesis Tracing (PMHT ethod has been developed by Streit and Luginbuhl in order to reduce the drawbacs of strong assignents. The PMHT ethod is presented here in a general ixture densities perspective. The Expectation-Maxiization (EM algorith is the basic ingredient for estiating ixture paraeters. This approach is then extended and applied to ultitarget tracing for nonlinear easureent odels in the passive sonar perspective. Manuscript received February 15, 1996; revised Septeber 9, IEEE Log No. T-AES/33/4/6848. Authors addresses H. Gauvrit and J. P. Le Cadre, IRISA/CNRS, Capus de Beaulieu, 3542 Rennes Cedex, France; C. Jauffret, GESSY, Avenue G. Popidou, B.P. 56, La Valette du Var, France. Multitarget tracing is concerned with states estiation of an unnown nuber of targets, in a surveillance region. Available easureents ay have originated fro the clutter or the targets of interest if detected. Due to this uncertainty, ultitarget tracing does not correspond to a traditional estiation proble. Two different probles have to be solved jointly data-association and estiation. Extension of standard estiation algoriths as Nearest Neighbor ethods generally provide poor results in dense clutter environent. In these algoriths, the prediction is based upon the easureents which lie in a suitable neighborhood of the predicted esureents. So, these ethods are essentially focused on estiation. On the contrary, the probabilistic data association filter (PDAF and its extension to ultiple targets, the joint PDAF (JPDAF, rely on an assignent step through the probability that a easureent originates fro a target. So, it is worth stressing that the fundaental difficulty in ultitarget tracing lies in data-association. Since the id-sixties, this subject has attracted uch attention especially because perforance of the algoriths used to solve the proble conditions greatly the overall perforance of the SONAR or RADAR syste. A natural fraewor consists to face the cobinatorial coplexity of the data-association proble. Morefield [1] first, and ore recently with the arrival of increasingly powerful coputers, Pattipati, et al. [13], Castañon [3], Poore, et al. [14], forulate this proble as a ultidiensional assignent one. This proble is nown to be NP-hard, that is to say that the coplexity grows exponentially with the nuber of scans and easureents. That is why these ethods were not intensively exploited at the beginning and soe alternatives were proposed. In 1979, Reid [16] presented a real-tie algorith, the Multiple Hypothesis Tracer (MHT in which easureents received at a scan are assigned to initialized targets, new targets or false alars. Pruning and gating techniques are used to retain the ost liely hypotheses and so liit their nuber. The ain ris is to eliinate correct easureent sequences and it is all the ore liely since the ore interesting sources ay be wea and fluctuating. Another way to treat this proble is to consider data-association fro a probabilistic viewpoint in order to copute the lielihood function. But the uncertainty in the easureents and the nuber of estiated paraeters ae it difficult to copute directly. Multitarget tracing can be viewed as an estiation proble but with partial observations. If the assignent vector of easureents to targets is available, it is an easy tas to calculate the lielihood function for each target. Then, ultitarget tracing becoes a traditional estiation proble. In this way, we can apply the Expectation-Maxiization (EM algorith to find the axiu lielihood (ML estiator. Avitzour [1] pioneered the use of this algorith for ultitarget tracing. In their seinal papers, Streit and Luginbuhl 1 [18, 19] gave a ore rigorous application of the EM algorith and proposed a batch Maxiu A Posteriori /97/$1. c 1997 IEEE 1 For the sequel, we refer to Streit and Luginbuhl as Streit, et al IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 33, NO. 4 OCTOBER 1997

2 (MAP algorith, called the Probabilistic Multiple Hypothesis Tracing (PMHT, which coupled EM iterations and Kalan filtering in the case of linear easureents. Based upon the duality of ultitarget tracing, the easureent assignents are odeled as rando variables which ust be estiated jointly with target states. This probabilistic approach to the proble is original and does not require any enueration of easureent to target associations. In fact, by avoiding the enueration proble, the probabilistic approach to the proble is not NP-hard. The approach used is quite siilar in its spirit to the one of Streit, et al. [19]. In this wor, a general forulation of ultitarget tracing in the fraewor of an incoplete data proble is proposed. First, this incoplete data proble is applied to finite ixture densities. The easureent-to-target assignent vector is naturally interpreted as the issing inforation. Thus, we define the couple of easureents and assignents as the coplete data (Section IV. Inclusion of data association in estiation is done through the extended state vector, coposed of target states and assignent probabilities. We develop two classes of algoriths (Section V one in the case of the ML and the other in the case of the MAP. For the latter, the general ixture fraewor allow us to retrieve the results obtained by Streit, et al. [19]. Then, we applyinsectionvi,themlapproachtothepassive SONAR for which, by definition, we ust face nonlinear easureent equations. The axiization procedures aresolvedbyeansofaodifiednewtonalgorith using the Levenberg-Marquardt ethod [21]. In the siulations, easureents were generated through the classical beaforing in order to tae into account trac coalescence in data association. The results deonstrate the good ability of the algorith to separate tracs and they bring out that ultitarget tracing perforance is closely related to signal processing since estiation of powerful target states ay be severely deteriorated due to biased easureents provided by the classical beaforing in crossing area. This wor is organized as follows. In Section II a general presentation of ultitarget tracing is provided, followed in Section III by a review of the assuptions ade in classical ultitarget tracing approaches in coparison with the assuptions retained in the probabilistic approach. Section IV deals with a general presentation of the EM algorith and its application to finite ixture densities. Results of this section are then extended to ultitarget tracing in Section V. The ML approach is applied in Section VI to state estiation of nonaneuvering targets for a narrowband passive SONAR. We show finally in Section VII that these results can be extended to the ultisensor data association proble. The following standard notations are used throughout the paper. Calligraphic letters indicate sets of batch length; upper case letters are used to denote sets for a given tie; lower case letters indicate vectors; Z denotes the easureents; denotes target states; K denotes the assignent vector; denotes the assignent probabilities; O denotes the paraeter vector; T is a trac; P is a partition of the easureents into tracs; N ( denotes the ultivariate Gaussian probability density function. II. GENERAL PROBLEM OF MULTITARGET TRACKING, NOTATIONS The purpose of this section is to present a general fraewor for ultitarget tracing in order to copare the different alternatives with the probabilistic one. Based on the ideas of Streit, et al., a general forulation is provided and extended by adding trac and partition definitions. This definition of the proble is quite siilartothatofmori,etal.[11]whereageneraland atheatical forulation of the proble is presented. In their approach, a very general ultisensor ultitarget tracing proble was considered. Here, the forulation of our proble is restricted to a single sensor and a batch forulation of the ultitarget tracing proble is considered. Let Z be the set of cuulative easureents and the set of cuulative state vector, both of length T ((1,,(T Z (Z(1,,Z(T For this batch length, we consider a set of M odels of targets oving in the surveillance region. This nuber is not restrictive and one odel can incorporate false alars. Target otions are odeled by Marov process whichcanbeexpressedinageneralwayfordiscrete tie as follows x s (tf s [x s (t 1,v s (t], t 1,,T where v s (t is a white Gaussian noise with covariance atrix Q s (t and the subscript s indicates the odel index. The associated easureent equation is given below z(th s [x s (t,n s (t], t 1,,T (1 where n s (t is the easureent Gaussian white noise with covariance atrix N s (t andz(t is the easureent which originates fro odel s. In the linear Gaussian case, these equations result in the classical ones ½ xs (t s (t,t 1x s (t 1 + v s (t, t 1,,T z(th s (tx s (t+w s (t, t 1,,T (2 At each tie t, anewvectorofeasureentsis received Z(t(z 1 (t,,z t (t Its size t varies in tie due to false alars, nondetected targets or trac initalizations. The corresponding state vector is constituted of the states of the M odels of targets at tie t. Itisdenoted (t(x 1 (t,,x M (t In order to tae into account the uncertainty in easureent origin, we introduce an assignent vector. The concatenated assignent vector is noted K and defined by K (K(1,,K(T GAUVRIT ET AL. FORMULATION OF MULTITARGET TRACKING AS AN INCOMPLETE DATA PROBLEM 1243

3 Each eleent K(t expresses an assignent hypothesis at tie t K(t( 1 (t,, t (t where j (ts indicates that target s produces easureent j at tie t. AtracT i is thus defined as a set of easureents assigned to the sae odel i T i fz j (t j j (ti, 1 t Tg, 1<i M In this definition, a trac ay contain issed detections (detection probability less than unity. As a odel is allocated to false alars, this definition includes false-alars trac. A partition of the easureents into tracs is defined as the set of nonepty tracs P ft i jt i 6Øg To each partition corresponds a data-association hypothesis of the eaureents Z. This notion of partition [1] iplies an exhaustive enueration of all the easureent to target assignents. Thus, data-association consists in finding the ost probable partition. It is obvious that if we consider a recursive hypothesis generation [16], their nuber will grow exponentially with tie. So, pruning and gating are used to reduce the coputational burden with a ris of eliinating soe correct hypotheses. The probabilistic approach proposed by Streit, et al. avoids this enueration by assigning all the easureents to each target in a probabilistic sense. Although any different approaches were proposed, they use a coon foralis. Consider for exaple two different approaches the JPDAF and the MHT. We do not detail these algoriths but just show how they can be expressed in ter of our definitions. In the JPDAF, hypotheses are built for the current easureent vector. An hypothesis is defined as the event associated with the corresponding assignent vector. The ajor ai of gating is to liit their nuber to the ost liely ones. On the other hand, the probabilities of these hypotheses [6] are calculated independently of gating. Suppose a new set of easureents be received at tie n, Z(n, the probability of the event associated with the assignent vector, K(n stands as follows 2 p(k(n j Z n 1 c p(z(n j K(n,Zn 1 p(k(n j Z n 1 where c represents the noralization constant. In this writing, we confused the assignent vector with the associated event. The first ter in the right-hand side identifies with the probability density function (pdf of the current scan conditionned on the assignent vector and the past easureents. The last ter of this side is the prior probability of such an event. It depends on the target detection probability and the false-alar density. By suing all the events for which easureent j originates fro target s, the event probabillity p( j (ns j Z n is calculated. These probabilities are used to update Kalan filters. 2 We adopt traditional notations for the cuulative data sets up to tie n, denoted Z n. We note the tracs built up to tie n, T i (n, and the corresponding partition P(n. In MHT, when a new set of easureents is received at tie n, each of the previously established hypotheses contained in the partition P(n 1, are extended by adding the current hypothesis associated with the assignent vector K(n to for a new partition P(n (P(n 1,K(n [16]. The probability of such a partition is coputed recursively p(p(n j Z n 1 c p(z(n jp(n 1,K(n,Zn 1 p(k(n jp(n 1,Z n 1 p(p(n 1 j Z n 1 where we identify the first two ters with those of the JPDAF and the last one with the probability of the past easureents. Conversely, in the approach of Streit, et al., partitionning is avoided because easureents are not assigned to particular targets but to all the targets. The assignent vector is considered as a rando variable. Streit, et al. defines a paraeter vector called the observer state O with continous and discrete coponents corresponding to the targets states and the assignent vector, respectively. This idea is original and powerful in the sense that data-association is basically included in the estiation proble. However in ultitarget tracing, the issing data are the assignent vectors. So, they cannot be included directly in the estiation vector. Instead, the assignent probability of a easureent to a odel is added to the target state vector. Let denote this new vector ( (1,, (T and (t(¼ 1 (t,,¼ M (t where the notation ¼ (t indicates the probability that a easureent originates fro the odel. In other words, this probability is independent of the easureent, i.e., ¼ (tp( j (t, for all j 1,, t Finally, let us define the paraeter vector as III. O (, HYPOTHESES IN MULTITARGET TRACKING In the previous section the general concepts of ultitarget tracing were introduced. We now focus our attention on the constraints associated with the assignent vector and the related statistical assuptions. In traditional approaches [6, 1, 16] the following holds true. 1 One easureent originates fro one target or the clutter (no erged easureent [11] which eans that the association is exclusive and exhaustive, i.e., p[ T i Z and T i \T j Ø, i1 i 6 j, i 1,,p and j 1,,p with p M the nuber of tracs in the partition P IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 33, NO. 4 OCTOBER 1997

4 2 A trac produces at ost one easureent (no split easureent [11] which underscores the dependence in assigning easureents inside a scan; suppose odel M is allocated to false alars, this assuption iplies t 1T, j 6 j, j 1,, t, j 1,, t 2f1,,Mg j (t j (t 6 The first assuption results in the following constraint on the assignent probabilities ¼ (t1 (3 1 In PMHT, only this assuption is specified, the second one is ignored so that easureents ay have the sae origin. Furtherore, it is assued that the coponents of the assignent vector K(t are independent. Consequently, the probability of the associated event is Y t p(k(t p( j (t Actually, this assuption is crucial for the derivation of the EM algorith. It corresponds to practical cases as in ultiple frequency line tracing. On the contrary, in classical approaches, the constraints on the assignent vector ipose on the calculation of the joint probability of such events. But, inside a scan, they consider easureents independent. This latter assuption is also retained in our approach. Furtherore, we assue that the assignent vectors are independent of target states at each tie t, and the target tracs are independent of each other. IV. EM ALGORITHM FOR MITURE DENSITIES This section is devoted to a brief presentation of the EM algorith and its application to ixture densities. The reference paper is [4]. Soe interesting additional inforation can be found in [9, 23] for the convergence criteria, in [17] for ixture densities, and in [22] for application to exponential failies. EM algoriths are used in any applications to solve incoplete data probles. Application of the EM algorith to the estiation of ixture paraeters is considered in this section. Using a siilar foralis, we show in the next section that the PMHT equations can be obtained in this way. A. Brief Review of EM Algorith Consider two spaces and Y andaappingfro to Y and the set (yfx j x 2 and y(xyg Let us assue that the pdfs of x and y are paraeterized by Á. They are denoted f(x j Á andg Á respectively, where Á (' 1,,' d 2 ½ R d The proble consists in estiating Á thans to the ML basedontheobservationofy. Westressthatx is not directly observable. This proble is well nown as incoplete data proble. Thereafter, we refer to x as the coplete data and to y as the observed incoplete data. Finally, we introduce the conditional density of x given y and Á, naely f(x j Á h(x j y,á g Á (4 This density represents the pdf of the issing data conditioned on the observed data and the paraeter vector. It plays an iportant part in the EM algorith. Since the density of the coplete data is unnown, it is estiated through the expectation of the pdf of the issing data (5. EM algorith is a ethod to find this axiu given an observation y. Each iteration coprises an expectation step (E-step and a axiization step (M-step. Denote Á i the paraeter estiated during the (ith iteration, the updated paraeter Á i+1 at the (i +1th iteration is obtained via the following (EM recursion. 1 E-step Copute the expectation defined lie this where Q(Á j Á i Eflog[f(x j Á] j y,á i g L(Á+H(Á j Á i (5 L(Álog[g Á] H(Á j Á i Eflog[h(x j y,á] j y,á i g 2 M-step Find Á which axiizes Q(Á j Á i Á i+1 argaxq(á j Á i Á Unnowing the lielihood of the coplete data, its expectation (E-step is calculated and axiized at the current step based upon the previous value of Á and the observation y. Using Jensen s inequalities, it can be shown that the log-lielihood L(Á increases at each iteration [4]. B. Mixture Densities The proble of estiating paraeters fro ixture densities has been the subject of an iportant literature. A brief review of different approaches is presented in the article of Redner, et al. [17] (other references can be found there. Paraeter estiation of ixture densities ay be achieved by eans of the EM algorith. Following the guidelines of Redner and Waler, the EM calculations is now detailed. Their utility for deriving the PMHT equations is deonstrated in the next section. As previously, denotes the coplete data vector and Y the incoplete data vector. Suppose the set of N independent data be observed, Y fy j g N.Each easureent y j belongs to a faily of paraetric density functions with probabilities f¼ i g i1 p j Á ¼ i p i j ' i i1 GAUVRIT ET AL. FORMULATION OF MULTITARGET TRACKING AS AN INCOMPLETE DATA PROBLEM 1245

5 where f¼ i g i1 and ¼ i 1 Each p i is a pdf paraetrized by ' i.denoteá the paraeter vector, Á (¼ 1,,¼,' 1,,'. Using the independence assuption for the easureents, the lielihood function and the log-lielihood express as follows NY g(y j Á ¼ i p i j ' i L(Á i1 i1 " N # log ¼ i p i j ' i To derive EM iterations, we need to introduce the density function of the coplete data and the density function of the issing data given the observed data. Let us define the coplete data vector as fx j g,,n where each coplete data x j is defined by x j, j and where the issing data j taes value in f1,,g, indicating which density the observed data y j coes fro. Notice that is a ixed continuous and discrete rando variable. The probability ass function of the issing data vector, K f j g,,n,is p(k j Á Eleentary calculations yield i1 NY ¼ j f( j Áp(Y j K,Áp(K j Á NY ¼ j p j j ' j (6 The probability ass function of the issing data conditioned on the observed data and the paraeter vector is easily deduced based on (4 and (6 h( j Y,Áp(K j Y,Á NY ¼ j p j j ' j (7 p j Á 1 E-step Suppose that the paraeters Á i have been estiated at iteration (i. To update the estiation, we ust calculate the following expectation Q(Á j Á i Eflog[f( j Á] j Y,Á i g The expectation Q(Á j Á i ay thus be deduced using (6 and (7 and stands as follows Q(Á j Á i log[f( j Á]p(K j Y,Á i ( N log[¼ j p j j ' j ] 1 1 N 1 NY ¼ i p (y j j j j ' i j (8 p j Á i j 1 Using straightforward calculations the following result (see Appendix A is obtained " N # ¼ i Q(Á j Á i p j 'i log[¼ p j Á i ] N log[p j ' ] ¼i p j 'i p j Á i REMARKS. 1 Notice that (9 is decoposed in two ters each one containing only one ind of paraeters. This attractive expression allows considerable siplification of the axiization step. 2 Moreover, suppose paraeters f' g 1 utually independent, axiization with respect to these paraeters is again decoposed in axiizations. 3 Finally, as log[¼ 1 ],,log[¼ ] appears linearly in the first ter in (9, an explicit solution to the first axiization can be deterined. 2 M-step Assaidabove,M-stepissolvedby axiizing (9 with respect to f¼ g 1,, on the one hand, and f' g 1,, on the other. Thus, the first axiization stands as " N # ¼ i Maxiize g(¼ p j 'i log[¼ p(y 1 j j Á i ] (1 subject to ¼ 1 1 Foring the Lagrangian L(¼, g(¼ + a necessary condition is The probability ¼ i+1 ¼ i+1 Ã 1 r ¼ L(¼,! ¼ 1 is straightforwardly deduced 1 N ¼ i p j 'i p j Á i where the Lagrange ultiplier is deterined by using the constraint (3. Since p j Á i ¼ i p s s j 'i s s1 eleentary calculation yields N Then, the new probability is updated at iteration (i +1 according to ¼ i+1 1 N j, (11 N ( IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 33, NO. 4 OCTOBER 1997

6 where ¼i p j 'i j, (12 p j Á i We have considered the application of the EM ethod to the estiation of ixture paraeters. This approach is now extended to ultitarget tracing. In a second tie, the second ter of (9 is axiized with respect to f' g 1. Let us recall the expression to axiize 1 N log[p j ' ] ¼i p j 'i p j Á i Since paraeters ' are utually independent in the previous expression, the paraeter vector ' is updated by ' i+1 2 Argax ' N log[p j ' ] j,, 8 1 (13 The algorith then taes the following for 1 update of ¼ ¼ i+1 1 N j, N where ¼i p j 'i j, ; p j Á i 2 update of ' according to N ' i+1 2 Argax log[p j ' ], j, ' 8 1 Each weight j, (12 corresponds to the posterior probability that easureent y j originates fro the th density given the current estiation Á i. Equation (9 taes advantage of all the inforation available at current iteration. In soe applications, axiization with respect to ' adits analytic updating solutions. Consider for exaple the case where p ' is a Gaussian density 1 p ' (2¼ n2 (det 12 e 12(y ¹ T ' (¹, 1 (y ¹, where y 2 R n, ¹ 2 R n and is an n n positive defined atrix. Then, the updated expression of ' i+1 at iteration (i +1 is given below P N y ¼ i p j 'i 1 p(y ¹ i+1 j j Á i P N ¼ i p j 'i, 1 p j Á i P N (y 1 ¹i+1 (y ¹ i+1 T ¼i p j 'i p(y i+1 j j Á i P N ¼ i p j 'i 1 p j Á i V. EM ALGORITHM APPLIED TO MULTITARGET TRACKING A batch algorith as in [19] is considered. A set of easureents Z is available. The assignent vector is not observable. So, we consider quite naturally that Z is the incoplete data set and that (Z,K is the coplete data set. Let us now define the different densities needed to derive EM algorith. Firstly, The probability ass function of the issing data K is now defined as 3 p(kjo TY Y t ¼ j (t (14 Then, the lielihood of the coplete data set is given below by eans of Bayes forula and independence assuptions 4 TY p(z,kjo p(z(t,k(t j O(t TY Y t p(z j (t j x j (t¼ j (t (15 Next, the lielihood functional of the incoplete data stands as follows TY p(z jo p(z(t j (t, (t TY Y t p(z j (t j O(t TY Y t p(z j (t j x (t¼ (t (16 1 Expression (16 is deduced fro the independence assuption about the assignent vector coponents at the current scan. Finally, a generalization of (7 to ultitarget tracing is p(kjz,o p(z,kjo TY Y t p( p(z jo j (t j z j (t,o(t (17 The algorith consists not only in estiating the targets states but also the assignent probabilities. A. E-Step Denote O the estiated vector at the previous iteration. This step is devoted to calculate the expectation 3 O (,, see p Independence fro scan to scan and fro assignent to assignent. GAUVRIT ET AL. FORMULATION OF MULTITARGET TRACKING AS AN INCOMPLETE DATA PROBLEM 1247

7 of the log-lielihood of coplete data based upon the nowledge of the paraeters at previous step and easureents. By analogy with (8, we consider the function Q(OjO Q(OjO Eflog[p(Z,KjO] jz,o g K log[p(z,kjo]p(kjz,o (18 P This calculation is an extension of (8. Denote nb T. The su coprises M nb t ters. Substituting (15 and (17 in (18, the independence assuption of K(t yields ( T t Q(OjO log[p(z j (t j x j (t¼ j (t] K ( T YY t p( j (t j z j (t,o (t Siplifying the sus, we obtain Q(OjO T t + j (t1 T t log[¼ j (t]w j,j (t j (t1 log[p(z j (t j x j (t]w j,j (t As j (t taes value in f1,,mg whatever t 1,,T and j 1,, t, we can invert sus in (19, yielding Q(OjO 1 + T 1 " t # w j, (t log[¼ (t] T t log[p(z j (t j x (t]w j, (t (19 (2 The rears fro the previous section still reain valid. The independence assuption of targets allow us to solve M axiizations, each coupled thans to the weighting w j, (t ¼ p(z (t j j x (t p(z j (t j O (t B. M-Step The first axiization proble reverts to T axiizations Maxiize for all t, g( (t subject to ¼ (t1 1 t log[¼ (t]w j, (t 1 (21 Solution to each axiization is given by (11 ¼ (t 1 t t w j, (t (22 The second axiization proble with respect to is equivalent to (13 generalized for all t 2f1,,Tg. Hence, for all t 1,,T and 1,,M, x (t 2 Argax x t log[p(z j (t j x (t]w j, (t (23 This last expression does not include tracing in states estiation. If we suppose that targets obey deterinist trajectories (see Section VI, state paraeters reduce to their initial states, denoted (x 1,,x 2,,,x M,. So the axiization step becoes for all 1,, M, x, 2 Argax x, T t log[p(z j (t j x, ]w j, (t C. EM in the Case of Maxiu A Posteriori Throughout the discussion, we only deal with theml.aneasyextensioncanbeintroducedtotae advantage of nowledge of prior probability relative to the paraeters. We show that a MAP derivation of the algorith corresponds to the PMHT of Streit, et al. which results in Kalan filtering in the linear Gaussian case [19]. Let us denote P(O log[p(, ]. Suppose assignent probabilities don t have prior probabilities as in [19], and the different odels follow a Marov process 5 P(O log[p(, ] log " p(( 1 # TY p((t j (t 1 " M # Y MY TY log p(x ( p(x (t j x (t 1 log[p(x (] (24 T log[p(x (t j x (t 1] (25 The MAP can be straightforwardly ebedded into the EM fraewor. The E-step of the algorith is changed 5 For the sequel, only target odels are considered in prior probability. The odel devoted to false alars has been dropped since no ineatic paraeter has to be estiated. Of course, such a odel is taen into account in assignent probabilities (see Section VI IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 33, NO. 4 OCTOBER 1997

8 in [4] M(OjO Q(OjO +P(O (26 Substituting (2 and (25 in (26 leads to " T t # M(OjO w j, (t log[¼ (t] T t log[p(z j (t j x (t]w j, (t log[p(x (] 1 1 T log[p(x (t j x (t 1] The axiization with respect to and to is always decoposed in independent axiizations coupled with weightings w j, (t. Assignent probabilities are updated according to (22. The state vectors for next iteration are given by (x xt 2 argax 8 >< > T t log[p(z j (t j x (t]w j, (t +log[p(x (] + T log[p(x (t j x (t 1] 9 > >; for all 1,,M (27 An interesting case is the linear Gaussian Marov process described by (2. Instead of axiizing directly (27, consider the exponential of this expression [19] ( TY Y t p(x p(x (t j x (t 1 p(z j (t j x (t w j,(t, and notice that Y t p(z j (t j x (t wi+1 j, (t where Y t / N (z j (t j H x (t,( j, (t 1 R /N( z (t j H x (t,( t ¼ i+1 (t 1 R 1 t z (t t ¼ i+1 j, (t (tz (t j 1,,M This axiization can be solved by using the Kalan filtering where the easureent is replaced with the easureent centroid z (t with covariance atrix R defined as R R t ¼ i+1 (t, D. Conclusion In conclusion, based upon an initialization of the paraeters denoted O ( ( (, (, the new paraeters at step (i + 1 are updated iteratively according to the following. 1 Update of (x i+1 ¼ i+1 (t 1 t t j, (t j, (t¼i p(z j (t j xi (t p(z j (t j O i (t where for all t 1,,T and 1,,M (28 2 Update of For the MAP ( x i+1 (T ( TY 2 argax p(x ( p(x (t j x (t 1 Y t p(z j (t j x (t wi+1 j, (t For the ML for all x, 2 Argax x, 1,,M T t log[p(z j (t j x, ]w j, (t (29 for all 1,,M (3 Maxiizations with respect to depend on the application and require generally the use of iterative optiization algoriths. In the next section, ultitarget tracing is applied to passive SONAR. The corresponding axiization probles are detailed. VI. MULTITARGET TRACKING IN PASSIVE SONAR By considering easureent history coposed of Cartesian positions for targets oving in horizontal plane, ultitarget tracing is considerably siplified since observability is ensured and easureent equation is a linear function of target states. On the contrary, receiver aneuvers are required for ensuring observability in the bearings-only tracing context. Furtherore, target state estiation is uch ore sensitive because optiizations in the M-step are no longer linear. In this section, we assue that the observability conditions are fulfilled. A. Paraeter Estiation In this section, targets ove with constant velocity vectors. The paraeter vector is thus denoted O (, where indicates the M initial target states as in the GAUVRIT ET AL. FORMULATION OF MULTITARGET TRACKING AS AN INCOMPLETE DATA PROBLEM 1249

9 whichcanbeexpressedinavectorialforas with r x f (R 1 x A x T N 1 B x (34 A x cos (x,1 sin (x,1 ±tcos (x,1 ±tsin (x, Fig. 1. E-step of the previous section Scenario in TMA. O (x 1,,,x M,, (1,, (T We focus on the update of i+1 based upon O i and drop the subscript on the target odel since the axiization with respect to is decoposed into M axiizations. Denote the receiver state (Fig. 1 as and the target state x obs (t(r xo,r yo,v xo,v yo T x s (t(r xs,r ys,v xs,v ys T The relative state vector is thus defined as x(tx s (t x obs (t(r x,r y,v x,v y T The easureent equation (2 is replaced with where z (t+n(t (31 (t arctan μ rx (t r y (t In (31, n(t is a white Gaussian noise with variance ¾ 2 (t. Since targets are supposed to ove at constant velocity, a ML approach is used. The M-step of the algorith consists in coputing (24. Using (31, the functional f(z j x which has to be axiized for each odel is f(z j x T t log[p(z j (t j x ] j (t (32 T t 1 2¾ 2 (t (z j (t (x,t2 j (t+ (33 In (33, (x,t is the target aziuth at tie t for initial target vector x. Ters independent of x are dropped fro (33. Taing the partial derivative with respect to x yields the gradient vector B x cos (x,t sin (x,t T±tcos (x,t T±tsin (x,t P 1 (z j (1 (x 1,1wi+1 j (1 P T N diag(¾ 2 (t, R x diag(r(t,. (z j (T (x,twi+1 j t 1,,T t 1,,T (T C A In (34, R x is the atrix of the range sources at sapling tie (±t is the sapling period for inital target state x while N is the easureent variance atrix if we suppose that all easureents within a scan have the sae variance. Notice that the gradient of the expression we need to axiize in M-step differs fro the gradient of the classical MLE in bearings-only TMA [12] only by the easureent ter B x. In cluttered environent, it is replaced by a ixing proportion of all the easureents with their assignent probabilities that they originate fro the target of interest given the previous estiation. A solution to this axiization proble is obtained nuerically with a Gauss Newton algorith iproved by the Levenberg Marquadt ethod [5]. This ethod uses a trust region strategy. Since each M-step ay face wea estiability, this ethod is required to avoid unrealistic estiates. Each iteration of this algorith consists in coputing x l+1 x l [AT x l R 1 x l A T R 1 N 1 B x l x l x l N 1 R 1 A x l x l + ¹ x l Id] 1 where Id is the atrix identity and ¹ x l is the current value used to perturb the Hessian r 2 x f.notethat in evaluating r 2 x f,ters@ 2 (x,t@ 2 x have been dropped. Suarizing the previous results, the PMHT algorith cobines EM and Gauss Newton iterations. Inside each EM iteration, the algorith have to copute M axiizations which can be easily ipleented on parallel coputer since calculations of the weightings j (t can be coputed separately. B. Nuerical Results r x f T 1 ¾ 2 t (z j (t (x,t j (t Whatever the detection syste (SONAR or RADAR, the data association step is a high level one whose inputs are the outputs of low level processing including signal 125 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 33, NO. 4 OCTOBER 1997

10 Fig. 2. Initialized trajectories and corresponding aziuths (case 1. processing (filtering, discrete Fourier transfor (DFT and array processing (beaforing, predetection, etc.. In a narrowband passive SONAR, roughly speaing, the signal processing step is coposed as follows signals received fro a sensor array are filtered, then sapled and analyzed (e.g., spectral analysis. A classical beaforing is perfored to estiate instantaneous target paraeters (bearings, etc.. Array processing also includes a predetection step. Our ai is now to define a realistic sonar siulation. So, the basic level of our siulations is the vector fored fro a single DFT bin fro each sensor; this vector is often called the snapshot vector. It is denoted. An assuption, coon in the passive array processing context, asserts that this vector is zero-ean, Gaussian circular with covariance given by ¾ i D id i + b Id i1 where * indicates transposition and conjugation. In the above forula, denotes the steering D i vector of target i (M targets present in the surveillance region and, in the case of a linear array, 6 isgivenbythe classical forula D i [1,exp( 2i¼f s,,exp( 2i¼(n c 1f s ] T where f s stands for the spatial frequency (f s cos i2, intersensor distance d 2. The paraeter ¾ i represents the power of target i while b is the noise power. In classical beaforing the spatial density of the field ipinging the array in the direction is estiated by the value of the following quadratic for P( D ˆ D In this expression, ˆ is the estiated cross-spectral atrix of the sensor outputs. If the averaged periodogra is considered, ˆ taes the following for ˆ 1 N 1 N t t t 6 Linear array constituted of n c equispaced sensors. where t is the tth vector of sensor DFT (N is the nuber of integrations. 1 Case 1 Pd 1and Pfa Forallthe siulations, the array processing paraeters tae the following values n c 32,N 125. In order to evaluate the ability of the data-association algorith for ultiple sources, false alars are not included at first. Extension of easureent generation to cluttered environent is straightforward. Finally, to initialize the paraeters of the algorith, we consider that each odel is equally probable and that no strategy is applied for target state.however,poor guess of the initial state ay have draatic effect on the estiated source states and assignent probabilities. To avoid this drawbac, we replace the direct axiization of the lielihood functional L(O log[p(z jo] by a sequence of interediate axiixations of functionals fl i (Og i1 N. In these functionals the easureent covariance is increased to tae into account ore easureents in the evaluation of the weightings j, (t. This idea of iniizing the influence of target states initiation have been previously derived in the probabilistic data association (PDA context [7]. More precisely, this sequence is defined as follows L i (O L(O,a i ¾wherethefa i g i1 N is a decreasing sequence such that a N 1. The state estiate obtained fro L i (O serves as the target state initiation for L i+1 (O. Fig. 2 presents a scenario for three targets with different signal to noise ratios. The first target is travelling fro position (2,25 T with speed (7 /s, 3 /s T, the second one fro (11,12 T with speed ( 3 /s,8/s T and the third one fro (2,1 T with speed ( 2 /s,9/s T. The signal-to-noise ratios are assued constant during the siulation, equal to ( 1 db, 15 db, 15 db T, respectively. Corresponding easureents are also shown in Fig. 2. Even if the target powers are quite siilar, it is worth stressing the absorption of wea targets by powerful ones, which results in a trac coalescence in crossing areas. Initialized target trajectories and corresponding aziuths (values see Table I are displayed in Fig. 2. The receiver trajectory is coposed of three legs to ensure syste observability. Its speed vector is (7 /s, /s T for odd leg and ( /s,7 /s T for even leg. The duration of the siulation is 18s and the sapling period is ±t 16 s. The results are presented GAUVRIT ET AL. FORMULATION OF MULTITARGET TRACKING AS AN INCOMPLETE DATA PROBLEM 1251

11 Fig. 3. Estiated trajectories and corresponding aziuths (case 1. Fig. 4. Estiated probabilities f¼ (tg T,1 M (case 1. TABLE I Initiation of Algorith in Figs. 3 and 4. Despite a poor initiation, the algorith converges to the global solution. Although local solutions or stationary points are possible, the use of the sequence of functions defined above reduces their nubers. Despite its siplicity, this siulation deonstrates the ability of the algorith to resolve data-association in difficult crossing areas. Soe interesting rears can be ade by considering siultaneously Figs. 3 and 4. REMARKS. 1 For tracs well separated, the probability that a easureent originates fro each odel is equally probable as it was expected and is verified up to t ' 11 n, and fro t ' 18 n to t ' 27 n. 2 Fro t ' 1 n to t ' 18 n, odel 2 and odel 3 share the sae easureents which are all assigned to the first one, thus assignent probability of odel 2 is equal to 2/3, for odel 3 to zero, and consequently for odel 1 to 1/3. The sae rear can be ade for odel2andodel1intheothercrossingareadefined for t ' 25 n up to the end of the scenario. Note in these areas, the instant when odels separate and the change in their corresponding probabilities. 3 Finally, as a conclusion of these rears, we can analyze the trac coalescence effects with respect to target powers. In the siulation, the ost powerful trac (odel 2 is poorly estiated in coparison with weaer ones (odel 1 and odel 3. In other words, state estiate of the ost powerful target is biased due to the bias of bearing estiates in crossing area. The probabilities of odel 1 and odel 3 are very wea in these areas, so their initial states are estiated based upon unbiased easureents. The less the difference between target powers is, the ore significant the bias is. Moreover, this phenoena is increased with the status of wea estiability. 2 Case 2 Multitarget Tracing in the Presence of False Detections In the previous section, the target s probability of detection, Pd, was assued equal to unity and no false alars were generated. In this section, the easureents that do not originate fro targets are independently and uniforly distributed in the easureent space, denoted U. The nuber of falseeasureentsineachscanfollowedapoisson distribution with paraeter, where was equal to the expected nuber of false alars per unit volue. The 1252 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 33, NO. 4 OCTOBER 1997

12 by Z P Us U ½ 1 exp 1 ¾ (2¼¾ ¾ (z (x 2,t2 dz Fig. 5. Siulated easureents (case 2. sae easureent vector was used in this section with a target detection probability fixed to Pd 8 for all the targets during the batch. Although this siulation does not integrate the detection step, it gives a good approxiation of reality. Fig. 5 depicts the siulated easureents for the sae scenario as in the previous section. Let us now describe the odifications of the algorith to include false alars. Suppose the (M 1 first odels describe target odels and the Mth one is devoted to false alars. Since no ineatic paraeter has to be estiated for such a odel, the M-step of the algorith always consists in the sae nuber of optiizations. Moreover, for each target the sae functional has to be axiized. It is straightforward to show that the odification of the algorith only lies in the evaluation of the assignent probabilities. The conditional probability density of a easureent, given the previous paraeter estiate and the odel fro which it is originated, is given below p(z j (t j x (t 8 >< > ½ ¾ 1 (2¼¾ 2 12 exp 1 2¾ 2 (z j (t (x,t2, 1 <M 1, M U Thus, the a posteriori probabilities that a easureent originates fro a target conditioned on the previous pareeter estiates is as follows (35 For the given scenario, initiation of the algorith is presented in Fig. 6 and the estiated trajectories in Fig. 7. This siulation deonstrates that the algorith is very good at solving the data association even in densely cluttered environent. Although the global solution is reached in this exaple, the algorith ay converge to local solutions or stationary points of the lielihood function. It is worth stressing that poor initializations can lead to poor data-association estiates, and thus, poor target state estiates. VII. ETENSION TO MULTISENSOR DATA ASSOCIATION Fro previous sections, the data association proble has focussed on teporal association of easureents. It is an easy tas to extend the approach to spatial data association of easureents fro ultiple sensors. The purpose is to study if we favour the solution to the data association proble by providing inforations fro S sensors. We show that the incoplete data fraewor extend easily to ultisensor tracing. For that purpose we introduce new notations ² Suppose we have S sensors. ² Z fz s g s1s where Z s denote the set of easureents received fro sensor s. ² denote always the state vector of the M odels. ² s,t is the nuber of easureents received by sensor s at tie t. ² K s is the assignent vector of easureents received by sensor s. ² denote assignent probabilities of the easureents to the M odels. We suppose that the set of easureents received fro different sensors are statistically independent. As a consequence, we consider the assignent vector of easureents Z s independent fro the assignent vector of Z s (s 6 s. Thus, the assignent proble becoes a 2-diensional data association proble w p(z (t j i+1 j, (t¼i j xi (t p(z j (t j O i (t j, (t 8 >< ¼M i (t (2¼¾2 12 U > ¼M i (t (2¼¾2 12 U ¼ i (t exp P U + P M 1 s1 + P M 1 s1 ½ 1 ¾ 2¾ (z (t (x 2 j,t2 ¼ i(t ½ s exp P Us ¼ i M (t (2¼¾2 12 U ¼ i(t s exp P Us ½ 1 2¾ 2 (z j (t (x s,t2 ¾, 1 <M 1 ¾, M 2¾ (z (t (x 2 j s,t2 (36 In (36, P Us is the noralization probability (i.e., the probability that the easureent originates fro the given odel and lies in the surveillance region defined (teporal and spatial. Moreover, we consider a centralized fusion where the assignent probabilities are estiated based on all the easureents received by the GAUVRIT ET AL. FORMULATION OF MULTITARGET TRACKING AS AN INCOMPLETE DATA PROBLEM 1253

13 Fig. 6. Initialized trajectories and corresponding aziuths (case 2. Fig. 7. Estiated trajectories and corresponding aziuths (case 2. centralized syste. Fro this viewpoint, the necessary lielihoods are redefined so as to derive EM algorith. The lielihood of the coplete data expresses as p(z,kjop(z 1 Z S,K 1 K S jo SY p(z s jk s,op(k s jo (37 s1 SY TY Y s,t p(z is (t j x is (t¼ is (t (38 s1 i s1 Then, the lielihood of the incoplete data is obtained by extending (16 to S sensors p(z jo SY p(z s jo s1 SY TY Y s,t p(z is (t j O(t s1 i s1 SY TY Y s,t p(z is (t j x (t¼ (t (39 s1 i s1 1 Finally, the density of the issing data is deduced fro (38 and (39 p(kjz,o p(z,kjo p(z jo SY TY Y s,t p( is (t j z is (t,o(t (4 s1 i s1 where p( is (t j z is (t,o(t p(z i s (t j x is (t¼ is (t P M 1 p(z i s (t j x (t¼ (t (41 Thus, the calulation of Q(OjO during E-step becoes Q(O jo log[p(z,kjo]p(kjz,o K ( S T s,t log[p(z is (t j x is (t¼ is (t] K ( S Y s1 i s1 TY Y s,t p( is (t j z is (t,o(t s1 i s1 The calculation derived in appendix extends straightforward fro the previous expressions since the decoposition required for this calculation reains valid. The siplified expression is therefore " T S s,t # Q(OjO w is,is (t log[¼ (t] T s1 i s1 S s,t log[p(z is (t j x is (t]w is,is (t s1 i s1 (42 In (42, w is, is (t is the weight of the easureent i s fro sensor s and for the odel is, or ore precisely the a posteriori probability to assign this easureent to 1254 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 33, NO. 4 OCTOBER 1997

14 Fig. 8. Measureents and initialized trajectories for two sensors. Fig. 9. Estiated trajectories and their corresponding aziuths for two sensors. odel is w is,is (t ¼ is p(z is (t j x is (t p(z is (t j O (t In the passive sonar context, if we suppose odel M is devoted to false-alars, we show that the updated probability of odel, ¼ (t, is obtained by suing the a posteriori probabilities to assign all the easureents within a scan and fro all the sensors to odel ¼ i+1 1 (t ( P S s1 s,t S s,t s1 i s1 i s, is (t (43 In (43, i s, is is calculated based on (36 where the easureent equation is replaced by the easureent equation of sensor s. Moreover, the axiization with respect to all the target state vectors is odified by adding the easureent equation for each sensor. We notice that the axiization of (42 can again be decoposed in a axiization for each odel. Since sources are oving at constant velocity, the function to axiize for a odel 2f1M 1g definedby(33is odified for S sensors in f(z j x T T S s,t log[p(z s,is (t j x ] i s (t (44 s1 i s1 S s,t 1 2¾s 2(t (z s,i s (t s(x,t 2 i s (t s1 i s1 + (45 The gradient of (45 expressed in a vectorial for is then r x f S (R 1 s,x A s,x T N 1 s B s,x (46 s1 where the ters in (46 for a sensor s have the sae eaningasin(34. We apply again a Gauss Newton algorith odified by the Levenberg Marquadt ethod to solve the axiization proble. Each iteration consists in coputing x l+1 x l " S S s1 s1 A T R 1 N 1 s,x l s,x l s R 1 s,x l A s,x l A T R 1 s,x l s,x l N 1 s B s,x l + ¹ x l Id # 1 In suary, we have proved that the ethod extends easily to ultisensor probles. More precisely, we have proposed a general fraewor to solve the difficult ultitarget ultisensor tracing proble. However, the nuerous siulations to which the algorith was subited underscore that adding new sensors do not contribute to the iproveent of the solution of the data association proble. The algorith converges regularly to soe stationary points or local solutions preventing an accurate estiation of target paraeters. We present a siulation on Figs. 8 and 9 where the algorith converges to the global solution. This scenario was built based on the scenario of the previous section in the case of one sensor. Here, the detection probability was equal tounityand15falsealarsweregeneratedineachscan. It brings to us the advantage of adding another sensor in ters of the quality of target paraeter estiation as soon as data association is correctly estiated. GAUVRIT ET AL. FORMULATION OF MULTITARGET TRACKING AS AN INCOMPLETE DATA PROBLEM 1255

15 VIII. CONCLUSION The ultitarget tracing has been considered in the natural fraewor of incoplete data probles. Based on the approach of Streit, et al., we have developed a ultitarget ultisensor tracing ethod for the passive sonar context. More precisely, the foralis of Streit, et al. has been extended by using the general fraewor of ixture paraeter estiation. The assignent vector is not incorporated into the paraeter vector but is interpreted as the issing data inforation. Thus, target state vector is estiated jointly with the probabilities that a easureent coes fro a target. The use of the EM algorith to copute the lielihood function or the MAP, allowed us to derive two different algoriths which appear very appealing because the global axiization with respect to all the target states is divided into M axiizations with respect to each target state, each coupled thans to the weightings w j (t. An extension of the teporal data association to the spatial data association has been investigated. This fraewor allows us to forulate the ultisensor ultitarget tracing proble. Siulation results for realistic scenarios have been considered. They deonstrate the good ability of the PMHT algorith to perfor ultitarget tracing in crossing areas. Such siulations perit us to study signal processing effects on ultitarget tracing perforance in an integrated way. Soe ore wor need to be pursued in the evaluation of the tracing perforance of the algorith and in the iproveent of its robustness. APPENDI A This Appendix is devoted to the calculation of the expectation of the E-step in ixture densities. Extension to ultitarget tracing is straightforward. At the E-step, we need to evaluate Q(Á j Á i 1 1 log[f(x j Á]p( j y,á i ( N log[¼ j p j j ' j ] N 1 NY ¼ i p (y j j j j ' i j p j Á i j 1 This expression can be siplified by considering the N sus on one of the eleent of the expression inter braces Q(Á j Á i N log[¼ j p j j ' j ] 1 1 N 1 NY ¼ i p (y j j j j ' i j j 1 p j Á i 8 N >< > j 1 log[¼ j p j j ' j ] ¼i j p j j ' i j p j Á i 1 1 j 1 1 j+1 1 NY ¼ i p (y j j j j ' i j p j Á i N 1 j 1 j 6j The previous expression coes fro a factorization of the ters in j. Using (7, the calculation of B j defined as B j 1 1 j 1 1 j+1 1 is an easy tas since B j 1 1 NY ¼ i p (y j j j (y j j ' i j p j Á i j 1 1 j+1 1 N 1 j 1 j 6j N 1 9 > >; (47 p( 1,, j 1, j+1,, N j y,á i 1 (48 and then by substituting (48 in (47 and inverting sus yields (9. ACKNOWLEDGMENTS The authors would lie to than R. L. Streit for all the insightful coents that helped in enhancing the quality of this paper. REFERENCES [1] Avitzour, D. (1992 A axiu lielihood approach to data association. IEEE Transactions on Aerospace and Electronics Systes, 28, 2 (Apr. 1992, [2] Bar-Shalo, Y. (Ed. (199 Multitarget-Multisensor Tracing Advanced Applications. Dedha, MA Artech House, 199. [3] Castañon, D. A. (1992 New assignent algoriths for data association. SPIE Proceedings, , Orlando, FL, Apr [4] Depster, A. P., Laird, N. M., and Rubin, D. B. (1977 Maxiu lielihood fro incoplete data via the EM algorith. Journal of Royal Statistical Society, SeriesB,39 (1977, [5] Dennis, J. E., Jr., and Schnabel, R. B. (1983 Nuerical Methods for Unconstrained Optiization and Nonlinear Equations. Englewood Cliffs, NJ Prentice Hall, [6] Fortann, T. E., Bar-Shalo, Y., and Scheffe, M. (1983 Sonar tracing of ultiple targets using joint probabilistic data association. IEEE Journal of Oceanic Research, OE-8 (July 1983, [7] Jauffret, C., and Bar-Shalo, Y. (199 Trac foration with bearing and frequency easureents in the presence of false detection. IEEE Transactions on Aerospace and Electronics Systes, 26, 6 (199, [8] Le Cadre, J. P., and Zugeyer, O. (1993 Teporal integration for array processing. Journal of the Acoustical Society of Aerica, 3 (Mar. 1993, [9] Louis, T. A. (1982 Finding the observed inforation atrix when using the EM algorith. Journal of the Royal Statistical Society, Ser.B,44, 2 (1982, IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 33, NO. 4 OCTOBER 1997

16 [1] [11] [12] [13] [14] [15] Morefield, C. L. (1977 Application of 1 integer prograing to ultitarget tracing probles. IEEE Transactions on Autoatic Control, AC-22, 3 (June 1977, Mori, S., Chong, C., Tse, E., and Wishner, R. P. (1986 Tracing and classifying ultiple targets without a priori identification. IEEE Transactions on Autoatic Control, AC-31, 5 (May 1986, Nardone, S. C., Lindgren, A. G., and Gong, K. F. (1984 Fundaental properties and perforance of conventional bearings-only target otion analysis. IEEE Transactions on Autoatic Control, AC-29, 9 (Sept. 1984, Pattipati, K. R., Deb, S., Bar-Shalo, Y., and Washburn, R. B. (1992 A new relaxation algorith and passive sensor data association. IEEE Transactions on Autoatic Control, AC-37, 2 (Feb. 1992, Poore, A. B., and Rijavec, N. (1991 Multi-target tracing and ulti-diensional assignent probles. In Proceedings of SPIE International Syposiu, Signal and Data Processing of Sall Targets 1991, Vol. 1481, Orlando, FL, Apr Rago, C., Willett, P., and Streit, R. (1995 A coparison of the JPDAF and PMHT tracing algoriths. In Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, 5 (May 9 12, 1995, [16] [17] [18] [19] [2] [21] [22] [23] Reid, D. B. (1979 An algorith for tracing ultiple targets. IEEE Transactions on Autoatic Control, AC-24, 6 (Dec. 1979, Redner, R. A., and Waler, H. F. (1984 Mixture densities, axiu lielihood and the EM algorith. Society for Industrial and Applied Matheatics, 26, 2 (Apr Streit, R. L., and Luginbuhl, T. E. (1993 A probabilistic ulti-hypothesis tracing algorith without enueration and pruning. In Proceedings of the Sixth Joint Service Data Fusion Syposiu, Laurel, MD, June 14 18, 1993, Streit, R. L., and Luginbuhl, T. E. (1994 Maxiu lielihood ethod for probabilistic ulti-hypothesis tracing. In Proceedings of SPIE International Syposiu, Signal and Data Processing of Sall Targets 1994, SPIE Proceedings Vol , Orlando, FL, Apr. 5 7, Streit, R. L., and Luginbuhl, T. E. (1994 Maxiu lielihood training of probabilistic neural networs. IEEE Transactions on Neural Networs, 5, 5 (Sept. 1994, Streit, R. L., and Luginbuhl, T. E. (1995 Probabilistic ulti-hypothesis tracing. Technical report 1428, Naval Undersea Warfare Center, Newport, RI, Feb. 15, Sundberg, R. (1976 An iterative ethod for solution of the lielihood equations for incoplete data fro exponential failies. Co. Statist. Siulation Coput., B5 (1976, Wu, C. F. (1983 On the convergence of the EM algorith. Annals of Statistics, 11, 1 (1983, Herve Gauvrit graduated fro the Ecole Supe rieure d Electronique de l Ouest, Angers, in 1992, and received his Diplo e d Etude Approfondie in signal processing fro the University of Rennes 1 in He joined IRISA in 1994 and is currently in the Ph.D. progra at the University of Rennes 1. His current research interests are in data association, estiation theory and its applications in ultitarget ultisensor tracing. Claude Jauffret was born on March 29, 1957, in Ollioules, France. He received the Diplo e d Etudes Approfondies in applied atheatics fro the Saint Charles University, Marseille, France in 1981 and the Diplo e D inge nieur fro the Ecole Nationale Supe rieure d Inforatique et de Mathe atiques Applique es de Grenoble in 1983, the title of docteur de l universite de Toulon in 1993 and the habilitation a diriger des recherches in 1996 fro the University of Toulon. Fro Nov to Sept. 1984, he wored on iage processing used in passive sonar systes at the GERDSM, Six-Fours Les Plages, France. Fro 1984 to 1988, he wored on target otion analysis (TMA probles at the GERDSM. A sabbatical year at the University of Connecticut, Storrs, allowed hi to wor on tracing probles in a cluttered environent in His research interests are ainly estiation and detection theory. He has published papers about observability and estiation in nonlinear systes and TMA. Since Septeber 1996, he is with the University of Toulon (France where he is a Professor. J. P. Le Cadre (M 93 received the M.S. degree in atheatics in 1977, the Doctorat de 3 ee cycle in 1982 and the Doctorat d Etat in 1987, both fro INPG, Grenoble, France. Fro 198 to 1989, he wored at the GERDSM (Groupe d Etudes et de Recherche en Detection Sous-Marines, a laboratory of the DCN (Direction des Constructions Navales, ainly on array processing. Since 1989, he has been with IRISA/CNRS, where he is a director of research. His interests have now oved toward topics lie syste analysis, detection, data association and operations research. He received (with O. Zugeyer the Eurasip Signal Processing best paper award in He is a eber of various societies of IEEE and of SIAM. GAUVRIT ET AL. FORMULATION OF MULTITARGET TRACKING AS AN INCOMPLETE DATA PROBLEM 1257