A Formulation of Multitarget Tracking as an Incomplete

A Formulation of MultitargetTracking as an IncompleteData Problem

H. GAUVRIT

J. P LE CADREIRISA/CNRSFrance

C. JAUFFRETDCN/Ingenierie/SudFrance

Traditional multihypothesis tracking methods rely upon

an enumeration of all the assignments of measurements to

tracks. Pruning and gating are used to retain only the most

likely hypotheses in order to drastically limit the set of feasible

associations. The main risk is to eliminate correct measurement

sequences.

The Probabilistic Multiple Hypothesis Tracking (PMHT)

method has been developed by Streit and Luginbuhl in order

to reduce the drawbacks of “strong” assignments. The PMHT

method is presented here in a general mixture densities

perspective. The Expectation-Maximization (EM) algorithm is

the basic ingredient for estimating mixture parameters. This

approach is then extended and applied to multitarget tracking for

nonlinear measurement models in the passive sonar perspective.

Manuscript received February 15, 1996; revised September 9, 1996.

IEEE Log No. T-AES/33/4/06848.

Authors’ addresses: H. Gauvrit and J. P. Le Cadre, IRISA/CNRS,Campus de Beaulieu, 35042 Rennes Cedex, France; C. Jauffret,GESSY, Avenue G. Pompidou, B.P. 56, 83 162 La Valette du Var,France.

0018-9251/97/$10.00 c° 1997 IEEE

I. INTRODUCTION

Multitarget tracking is concerned with statesestimation of an unknown number of targets, in asurveillance region. Available measurements mayhave originated from the clutter or the targets ofinterest if detected. Due to this uncertainty, multitargettracking does not correspond to a traditional estimationproblem. Two different problems have to be solvedjointly: data-association and estimation. Extension ofstandard estimation algorithms as Nearest Neighbormethods generally provide poor results in dense clutterenvironment. In these algorithms, the prediction isbased upon the measurements which lie in a suitableneighborhood of the predicted mesurements. So, thesemethods are essentially focused on estimation. Onthe contrary, the probabilistic data association filter(PDAF) and its extension to multiple targets, the jointPDAF (JPDAF), rely on an assignment step through theprobability that a measurement originates from a target.So, it is worth stressing that the fundamental difficulty inmultitarget tracking lies in data-association.

Since the mid-sixties, this subject has attractedmuch attention especially because performance ofthe algorithms used to solve the problem conditionsgreatly the overall performance of the SONAR orRADAR system. A natural framework consists to facethe combinatorial complexity of the data-associationproblem. Morefield [10] first, and more recently withthe arrival of increasingly powerful computers, Pattipati,et al. [13], Castanon [3], Poore, et al. [14], formulatethis problem as a multidimensional assignment one.This problem is known to be NP-hard, that is to say thatthe complexity grows exponentially with the numberof scans and measurements. That is why these methodswere not intensively exploited at the beginning and somealternatives were proposed. In 1979, Reid [16] presenteda real-time algorithm, the Multiple Hypothesis Tracker(MHT) in which measurements received at a scan areassigned to initialized targets, new targets or false alarms.Pruning and gating techniques are used to retain the mostlikely hypotheses and so limit their number. The mainrisk is to eliminate correct measurement sequences and itis all the more likely since the more interesting sourcesmay be weak and fluctuating.

Another way to treat this problem is to considerdata-association from a probabilistic viewpoint inorder to compute the likelihood function. But theuncertainty in the measurements and the number ofestimated parameters make it difficult to computedirectly. Multitarget tracking can be viewed as anestimation problem but with partial observations. Ifthe assignment vector of measurements to targets isavailable, it is an easy task to calculate the likelihoodfunction for each target. Then, multitarget trackingbecomes a traditional estimation problem. In this way, wecan apply the Expectation-Maximization (EM) algorithmto find the maximum likelihood (ML) estimator. Avitzour[1] pioneered the use of this algorithm for multitargettracking. In their seminal papers, Streit and Luginbuhl1

[18, 19] gave a more rigorous application of the EMalgorithm and proposed a batch Maximum A Posteriori

1For the sequel, we refer to Streit and Luginbuhl as Streit, et al.

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(MAP) algorithm, called the Probabilistic MultipleHypothesis Tracking (PMHT), which coupled EMiterations and Kalman filtering in the case of linearmeasurements. Based upon the duality of multitargettracking, the measurement assignments are modeled asrandom variables which must be estimated jointly withtarget states. This probabilistic approach to the problemis original and does not require any enumeration ofmeasurement to target associations. In fact, by avoidingthe enumeration problem, the probabilistic approach tothe problem is not NP-hard.

The approach used is quite similar in its spirit tothe one of Streit, et al. [19]. In this work, a generalformulation of multitarget tracking in the frameworkof an incomplete data problem is proposed. First, thisincomplete data problem is applied to finite mixturedensities. The measurement-to-target assignment vectoris naturally interpreted as the missing information.Thus, we define the couple of measurements andassignments as the complete data (Section IV). Inclusionof data association in estimation is done through theextended state vector, composed of target states andassignment probabilities. We develop two classes ofalgorithms (Section V): one in the case of the MLand the other in the case of the MAP. For the latter,the general mixture framework allow us to retrievethe results obtained by Streit, et al. [19]. Then, weapply in Section VI, the ML approach to the passiveSONAR for which, by definition, we must face nonlinearmeasurement equations. The maximization proceduresare solved by means of a modified Newton algorithmusing the Levenberg-Marquardt method [21]. In thesimulations, measurements were generated through theclassical beamforming in order to take into account trackcoalescence in data association. The results demonstratethe good ability of the algorithm to separate tracks andthey bring out that multitarget tracking performanceis closely related to signal processing since estimationof powerful target states may be severely deteriorateddue to biased measurements provided by the classicalbeamforming in crossing area. This work is organizedas follows. In Section II a general presentation ofmultitarget tracking is provided, followed in SectionIII by a review of the assumptions made in classicalmultitarget tracking approaches in comparison withthe assumptions retained in the probabilistic approach.Section IV deals with a general presentation of the EMalgorithm and its application to finite mixture densities.Results of this section are then extended to multitargettracking in Section V. The ML approach is applied inSection VI to state estimation of nonmaneuvering targetsfor a narrowband passive SONAR. We show finally inSection VII that these results can be extended to themultisensor data association problem.

The following standard notations are used throughoutthe paper. Calligraphic letters indicate sets of batchlength; upper case letters are used to denote sets fora given time; lower case letters indicate vectors; Zdenotes the measurements; X denotes target states; Kdenotes the assignment vector; ¦ denotes the assignmentprobabilities; O denotes the parameter vector; T is atrack; P is a partition of the measurements into tracks;N (¢) denotes the multivariate Gaussian probabilitydensity function.

II. GENERAL PROBLEM OF MULTITARGETTRACKING, NOTATIONS

The purpose of this section is to present a generalframework for multitarget tracking in order to comparethe different alternatives with the probabilistic one. Basedon the ideas of Streit, et al., a general formulation isprovided and extended by adding track and partitiondefinitions. This definition of the problem is quitesimilar to that of Mori, et al. [11] where a general andmathematical formulation of the problem is presented.In their approach, a very general multisensor multitargettracking problem was considered. Here, the formulationof our problem is restricted to a single sensor and abatch formulation of the multitarget tracking problemis considered.

Let Z be the set of cumulative measurements and Xthe set of cumulative state vector, both of length T:

X = (X(1), : : : ,X(T))

Z = (Z(1), : : : ,Z(T)):

For this batch length, we consider a set of M models oftargets moving in the surveillance region. This numberis not restrictive and one model can incorporate falsealarms. Target motions are modeled by Markov processwhich can be expressed in a general way for discretetime as follows:

xs(t) = fs[xs(t¡ 1),vs(t)], t= 1, : : : ,T

where vs(t) is a white Gaussian noise with covariancematrix Qs(t) and the subscript s indicates the modelindex.

The associated measurement equation is given below

z(t) = hs[xs(t),ns(t)], t= 1, : : : ,T (1)

where ns(t) is the measurement Gaussian white noisewith covariance matrix Ns(t) and z(t) is the measurementwhich originates from model s.

In the linear Gaussian case, these equations result inthe classical ones½

xs(t)= ©s(t, t¡ 1)xs(t¡ 1)+ vs(t), t = 1, : : : ,T

z(t)=Hs(t)xs(t)+ws(t), t = 1, : : : ,T:

(2)

At each time t, a new vector of measurements isreceived

Z(t) = (z1(t), : : : ,zmt (t)):

Its size mt varies in time due to false alarms, nondetectedtargets or track initalizations. The corresponding statevector is constituted of the states of the M models oftargets at time t. It is denoted

X(t) = (x1(t), : : : ,xM(t)):

In order to take into account the uncertainty inmeasurement origin, we introduce an assignment vector.The concatenated assignment vector is noted K anddefined by

K= (K(1), : : : ,K(T)):

GAUVRIT ET AL.: FORMULATION OF MULTITARGET TRACKING AS AN INCOMPLETE DATA PROBLEM 1243


Each element K(t) expresses an assignment hypothesis attime t:

K(t) = (k1(t), : : : ,kmt (t))

where kj(t) = s indicates that target s producesmeasurement j at time t.

A track Ti is thus defined as a set of measurementsassigned to the same model i:

Ti = fzj(t) j kj(t) = i, 1· t · Tg, 1< i ·M:

In this definition, a track may contain missed detections(detection probability less than unity). As a modelis allocated to false alarms, this definition includesfalse-alarms “track.”

A partition of the measurements into tracks is definedas the set of nonempty tracks:

P = fTi j Ti 6=Øg:

To each partition corresponds a data-associationhypothesis of the meaurements Z. This notion ofpartition [10] implies an exhaustive enumeration ofall the measurement to target assignments. Thus,data-association consists in finding the most probablepartition. It is obvious that if we consider a recursivehypothesis generation [16], their number will growexponentially with time. So, pruning and gatingare used to reduce the computational burden witha risk of eliminating some correct hypotheses. Theprobabilistic approach proposed by Streit, et al. avoidsthis enumeration by assigning all the measurements toeach target in a probabilistic sense.

Although many different approaches were proposed,they use a common formalism. Consider for example twodifferent approaches: the JPDAF and the MHT. We donot detail these algorithms but just show how they canbe expressed in term of our definitions. In the JPDAF,hypotheses are built for the current measurement vector.An hypothesis is defined as the event associated withthe corresponding assignment vector. The major aim ofgating is to limit their number to the most likely ones.On the other hand, the probabilities of these hypotheses[6] are calculated independently of gating.

Suppose a new set of measurements be received attime n, Z(n), the probability of the event associated withthe assignment vector, K(n) stands as follows:2

p(K(n) j Zn) = 1cp(Z(n) jK(n),Zn¡1)p(K(n) j Zn¡1)

where c represents the normalization constant.In this writing, we confused the assignment vector

with the associated event. The first term in the right-handside identifies with the probability density function (pdf)of the current scan conditionned on the assignmentvector and the past measurements. The last term ofthis side is the prior probability of such an event. Itdepends on the target detection probability and thefalse-alarm density. By summing all the events forwhich measurement j originates from target s, theevent probabillity p(kj(n) = s j Zn) is calculated. Theseprobabilities are used to update Kalman filters.

2We adopt traditional notations for the cumulative data sets up totime n, denoted Zn. We note the tracks built up to time n, Ti(n), andthe corresponding partition P(n).

In MHT, when a new set of measurements is receivedat time n, each of the previously established hypothesescontained in the partition P(n¡ 1), are extended byadding the current hypothesis associated with theassignment vector K(n) to form a new partition P(n) =(P(n¡ 1),K(n)) [16]. The probability of such a partitionis computed recursively:

p(P(n) j Zn) = 1cp(Z(n) j P(n¡ 1),K(n),Zn¡1)

£p(K(n) j P(n¡ 1),Zn¡1)

£p(P(n¡1) j Zn¡1)

where we identify the first two terms with those of theJPDAF and the last one with the probability of the pastmeasurements.

Conversely, in the approach of Streit, et al.,partitionning is avoided because measurements are notassigned to particular targets but to all the targets. Theassignment vector is considered as a random variable.Streit, et al. defines a parameter vector called theobserver state O with continous and discrete componentscorresponding to the targets states and the assignmentvector, respectively. This idea is original and powerful inthe sense that data-association is basically included in theestimation problem. However in multitarget tracking, themissing data are the assignment vectors. So, they cannotbe included directly in the estimation vector. Instead, theassignment probability of a measurement to a model isadded to the target state vector.

Let ¦ denote this new vector:

¦ = (¦(1), : : : ,¦(T)) and

¦(t) = (¼1(t), : : : ,¼M(t))

where the notation ¼m(t) indicates the probabilitythat a measurement originates from the model m. Inother words, this probability is independent of themeasurement, i.e.,

¼m(t) = p(kj(t) =m), for all j = 1, : : : ,mt:

Finally, let us define the parameter vector as

O ¢=(X ,¦):

III. HYPOTHESES IN MULTITARGET TRACKING

In the previous section the general concepts ofmultitarget tracking were introduced. We now focusour attention on the constraints associated with theassignment vector and the related statistical assumptions.In traditional approaches [6, 10, 16] the following holdstrue.

1) One measurement originates from one target orthe clutter (no merged measurement [11]) which meansthat the association is exclusive and exhaustive, i.e.,

p[i=1

Ti = Z and Ti \Tj =Ø,

i 6= j, i = 1, : : : ,p and j = 1, : : : ,p

with p·M the number of tracks in the partition P .



2) A track produces at most one measurement(no split measurement [11]) which underscores thedependence in assigning measurements inside a scan;suppose model M is allocated to false alarms, thisassumption implies:

t = 1 : : :T, j 6= j 0, j = 1, : : : ,mt,

j 0 = 1, : : : ,mt m 2 f1, : : : ,Mg

kj(t) =m) kj0(t) 6=m:

The first assumption results in the following constrainton the assignment probabilities:

MXm=1

¼m(t) = 1: (3)

In PMHT, only this assumption is specified, the secondone is ignored so that measurements may have the sameorigin. Furthermore, it is assumed that the componentsof the assignment vector K(t) are independent.Consequently, the probability of the associated event is

p(K(t)) =mtYj=1

p(kj(t)):

Actually, this assumption is crucial for the derivation ofthe EM algorithm. It corresponds to practical cases asin multiple frequency line tracking. On the contrary, inclassical approaches, the constraints on the assignmentvector impose on the calculation of the joint probabilityof such events. But, inside a scan, they considermeasurements independent. This latter assumption is alsoretained in our approach. Furthermore, we assume thatthe assignment vectors are independent of target states ateach time t, and the target tracks are independent of eachother.

IV. EM ALGORITHM FOR MIXTURE DENSITIES

This section is devoted to a brief presentation of theEM algorithm and its application to mixture densities.The reference paper is [4]. Some interesting additionalinformation can be found in [9, 23] for the convergencecriteria, in [17] for mixture densities, and in [22] forapplication to exponential families. EM algorithms areused in many applications to solve incomplete dataproblems. Application of the EM algorithm to theestimation of mixture parameters is considered in thissection. Using a similar formalism, we show in the nextsection that the PMHT equations can be obtained in thisway.

A. Brief Review of EM Algorithm

Consider two spaces X and Y and a mapping from Xto Y and the set

X (y) = fx j x 2 X and y(x) = yg:

Let us assume that the pdfs of x and y are parameterizedby Á. They are denoted f(x j Á) and g(y j Á) respectively,where

Á= ('1, : : : ,'d) 2 − ½ Rd:

The problem consists in estimating Á thanks to the MLbased on the observation of y. We stress that x is notdirectly observable. This problem is well known asincomplete data problem. Thereafter, we refer to x as thecomplete data and to y as the observed incomplete data.Finally, we introduce the conditional density of x given yand Á, namely:

h(x j y,Á) = f(x j Á)g(y j Á) : (4)

This density represents the pdf of the missing dataconditioned on the observed data and the parametervector. It plays an important part in the EM algorithm.Since the density of the complete data is unknown, itis estimated through the expectation of the pdf of themissing data (5).

EM algorithm is a method to find this maximumgiven an observation y. Each iteration comprises anexpectation step (E-step) and a maximization step(M-step). Denote Ái the parameter estimated during the(i)th iteration, the updated parameter Ái+1 at the (i+1)thiteration is obtained via the following (EM) recursion.

1) E-step: Compute the expectation defined like this

Q(Á j Ái) = Eflog[f(x j Á)] j y,Áig

= L(Á)+H(Á j Ái) (5)

where

L(Á) = log[g(y j Á)]

H(Á j Ái) = Eflog[h(x j y,Á)] j y,Áig:

2) M-step: Find Á which maximizes Q(Á j Ái)

Ái+1 = argmaxÁQ(Á j Ái):

Unknowing the likelihood of the complete data, itsexpectation (E-step) is calculated and maximized at thecurrent step based upon the previous value of Á andthe observation y. Using Jensen’s inequalities, it canbe shown that the log-likelihood L(Á) increases at eachiteration [4].

B. Mixture Densities

The problem of estimating parameters from mixturedensities has been the subject of an important literature.A brief review of different approaches is presented inthe article of Redner, et al. [17] (other references can befound there). Parameter estimation of mixture densitiesmay be achieved by means of the EM algorithm.Following the guidelines of Redner and Walker, the EMcalculations is now detailed. Their utility for deriving thePMHT equations is demonstrated in the next section.

As previously, X denotes the complete data vectorand Y the incomplete data vector. Suppose the set ofN independent data be observed, Y = fyjgj=1:::N . Eachmeasurement yj belongs to a family of parametric densityfunctions with probabilities f¼igi=1:::m:

p(yj j Á) =mXi=1

¼ipi(yj j 'i)



where

f¼i ¸ 0gmi=1 andmXi=1

¼i = 1:

Each pi is a pdf parametrized by 'i. Denote Á theparameter vector, Á= (¼1, : : : ,¼m,'1, : : : ,'m). Using theindependence assumption for the measurements, thelikelihood function and the log-likelihood express asfollows:

g(Y j Á) =NYj=1

mXi=1

¼ipi(yj j 'i)

L(Á) =NXj=1

log

"mXi=1

¼ipi(yj j 'i)#:

To derive EM iterations, we need to introduce thedensity function of the complete data and the densityfunction of the missing data given the observed data.Let us define the complete data vector as X = fxjgj=1,:::,Nwhere each complete data xj is defined by xj = (yj ,kj)and where the missing data kj takes value in f1, : : : ,mg,indicating which density the observed data yj comesfrom. Notice that X is a mixed continuous and discreterandom variable. The probability mass function of themissing data vector, K = fkjgj=1,:::,N , is

p(K j Á) =NYj=1

¼kj :

Elementary calculations yield

f(X j Á) = p(Y j K,Á)p(K j Á)

=NYj=1

¼kj pkj (yj j 'kj ): (6)

The probability mass function of the missing dataconditioned on the observed data and the parametervector is easily deduced based on (4) and (6):

h(X j Y,Á) = p(K j Y,Á) =NYj=1

¼kj pkj (yj j 'kj )p(yj j Á)

: (7)

1) E-step: Suppose that the parameters Ái have beenestimated at iteration (i). To update the estimation, wemust calculate the following expectation:

Q(Á j Ái) = Eflog[f(X j Á)] j Y,Áig:

The expectation Q(Á j Ái) may thus be deduced using (6)and (7) and stands as follows:

Q(Á j Ái) =Xk

log[f(X j Á)]p(K j Y,Ái)

=mX

k1=1

¢ ¢ ¢mX

kN=1

(NXj=1

log[¼kjpkj (yj j 'kj )])

£NYj0=1

¼ikj0pkj0(yj0 j 'ikj0 )

p(yj0 j Ái): (8)

Using straightforward calculations the following result(see Appendix A) is obtained

Q(Á j Ái) =mXk=1

"NXj=1

¼ikpk(yj j 'ik)p(yj j Ái)

#log[¼k]

+mXk=1

NXj=1

log[pk(yj j 'k)]¼ikpk(yj j 'ik)p(yj j Ái)

:

(9)

REMARKS.1) Notice that (9) is decomposed in two terms each

one containing only one kind of parameters. Thisattractive expression allows considerable simplificationof the maximization step.2) Moreover, suppose parameters f'kgk=1¢¢¢m mutually

independent, maximization with respect to theseparameters is again decomposed in m maximizations.3) Finally, as log[¼1], : : : , log[¼m] appears linearly

in the first term in (9), an explicit solution to the firstmaximization can be determined.

2) M-step: As said above, M-step is solved bymaximizing (9) with respect to f¼kgk=1,:::,m on the onehand, and f'kgk=1,¢¢¢ ,m on the other.

Thus, the first maximization stands as

Maximize g(¼) =mXk=1

"NXj=1


#log[¼k]

(10)

subject tomXk=1

¼k = 1:

Forming the Lagrangian

L(¼,¸) = g(¼)+¸

Ã1¡

mXk=1

¼k

!a necessary condition is

r¼L(¼,¸) = 0:

The probability ¼i+1k is straightforwardly deduced

¼i+1k =1¸

NXj=1


where the Lagrange multiplier ¸ is determined by usingthe constraint (3). Since

p(yj j Ái) =mXs=1

¼isps(yj j 'is)

elementary calculation yields

¸=N:

Then, the new probability is updated at iteration (i+1)according to

¼i+1k =1N

NXj=1

wi+1j,k (11)



where

wi+1j,k =¼ikpk(yj j 'ik)p(yj j Ái)

: (12)

In a second time, the second term of (9) ismaximized with respect to f'kgk=1¢¢¢m. Let us recall theexpression to maximize:

mXk=1

NXj=1

log[pk(yj j 'k)]¼ikpk(yj j 'ik)p(yj j Ái)

:

Since parameters 'k are mutually independent in theprevious expression, the parameter vector 'k is updatedby

'i+1k 2Argmax'k

NXj=1

log[pk(yj j 'k)]wi+1j,k ,

8 k = 1 ¢ ¢ ¢m: (13)

The algorithm then takes the following form:¯¯¯¯¯¯

1: update of ¼k :

¼i+1k =1N

NXj=1

wi+1j,k

where wi+1j,k =¼ikpk(yj j 'ik)p(yj j Ái)

;

2: update of 'k according to:

'i+1k 2Argmax'k

NXj=1

log[pk(yj j 'k)]wi+1j,k ,

8 k = 1 ¢ ¢ ¢m:

Each weight wi+1j,k (12) corresponds to the posteriorprobability that measurement yj originates from the kthdensity given the current estimation Ái. Equation (9)takes advantage of all the information available at currentiteration. In some applications, maximization with respectto 'k admits analytic updating solutions. Consider forexample the case where pk(y j 'k) is a Gaussian density:

pk(y j 'k) =1

(2¼)n=2(det§k)1=2e¡1=2(y¡¹k )

T§¡1k(y¡¹k ),

'k = (¹k,§k)

where y 2 Rn, ¹k 2 Rn and §k is an n£ n positive definedmatrix.

Then, the updated expression of 'i+1k at iteration(i+1) is given below¯

¯¯¯¯

¹i+1k =

PN

k=1 yk¼ikpk(yj j 'ik)p(yj j Ái)PN

k=1


,

§i+1k =

PN

k=1(yk ¡¹i+1k )(yk ¡¹i+1k )T¼ikpk(yj j 'ik)p(yj j Ái)PN

k=1

¼ikpk(yj j 'ik)p(yj j Ái

:

We have considered the application of the EMmethod to the estimation of mixture parameters. Thisapproach is now extended to multitarget tracking.

V. EM ALGORITHM APPLIED TO MULTITARGETTRACKING

A batch algorithm as in [19] is considered. A set ofmeasurements Z is available. The assignment vector isnot observable. So, we consider quite naturally that Z isthe incomplete data set and that (Z,K) is the completedata set. Let us now define the different densities neededto derive EM algorithm. Firstly, The probability massfunction of the missing data K is now defined as:3

p(K j O) =TYt=1

mtYj=1

¼kj (t): (14)

Then, the likelihood of the complete data set is givenbelow by means of Bayes’ formula and independenceassumptions:4

p(Z,K j O) =TYt=1

p(Z(t),K(t) jO(t))

=TYt=1

mtYj=1

p(zj(t) j xkj (t))¼kj (t): (15)

Next, the likelihood functional of the incomplete datastands as follows:

p(Z j O) =TYt=1

p(Z(t) j X(t),¦(t))

=TYt=1

mtYj=1

p(zj(t) jO(t))

=TYt=1

mtYj=1

MXm=1

p(zj(t) j xm(t))¼m(t): (16)

Expression (16) is deduced from the independenceassumption about the assignment vector components atthe current scan.

Finally, a generalization of (7) to multitarget trackingis

p(K j Z,O) = p(Z,K j O)p(Z j O) =

TYt=1

mtYj=1

p(kj(t) j zj(t),O(t)):

(17)

The algorithm consists not only in estimating the targetsstates but also the assignment probabilities.

A. E-Step

Denote O¤ the estimated vector at the previousiteration. This step is devoted to calculate the expectation

3O ¢=(X ,¦), see p. 7.

4Independence from scan to scan and from assignment toassignment.



of the log-likelihood of complete data based uponthe knowledge of the parameters at previous step andmeasurements. By analogy with (8), we consider thefunction Q(O j O¤):

Q(O j O¤) = Eflog[p(Z,K j O)] j Z,O¤g

=XK

log[p(Z,K j O)]p(K j Z,O¤):

(18)

This calculation is an extension of (8). Denote nb =PT

t=1mt. The sum comprises Mnb terms. Substituting (15)and (17) in (18), the independence assumption of K(t)yields

Q(O j O¤) =XK

(TXt=1

mtXj=1

log[p(zj(t) j xkj (t))¼kj (t)])

£(

TYt=1

mtYj=1

p(kj(t) j zj(t),O¤(t))

):

Simplifying the sums, we obtain

Q(O j O¤) =TXt=1

mtXj=1

MXkj (t)=1

log[¼kj (t)]wj,kj (t)

+TXt=1

mtXj=1

MXkj (t)=1

log[p(zj(t) j xkj (t))]wj,kj (t):

(19)

As kj(t) takes value in f1, : : : ,Mg whatever t= 1, : : : ,Tand j = 1, : : : ,mt, we can invert sums in (19), yielding

Q(O j O¤) =MXk=1

TXt=1

"mtXj=1

wj,k(t)

#log[¼k(t)]

+MXk=1

TXt=1

mtXj=1

log[p(zj(t) j xk(t))]wj,k(t):

(20)

The remarks from the previous section still remain valid.The independence assumption of targets allow us to solveM maximizations, each coupled thanks to the weighting

wj,k(t) =¼¤kp(zj(t) j x¤k(t))p(zj(t) jO¤(t))

:

B. M-Step

The first maximization problem reverts to Tmaximizations:

Maximize for all t, g(¦(t)) =MXk=1

mtXj=1

log[¼k(t)]wj,k(t)

(21)

subject toMXk=1

¼k(t) = 1:

Solution to each maximization is given by (11):

¼k(t) =1mt

mtXj=1

wj,k(t): (22)

The second maximization problem with respect toX is equivalent to (13) generalized for all t 2 f1, : : : ,Tg.Hence, for all t = 1, : : : ,T and k = 1, : : : ,M ,

xk(t) 2Argmaxxk

mtXj=1

log[p(zj(t) j xk(t))]wj,k(t): (23)

This last expression does not include tracking in statesestimation. If we suppose that targets obey deterministtrajectories (see Section VI), state parameters reduce totheir initial states, denoted X0 = (x1,0,x2,0, : : : ,xM,0). So themaximization step becomes: for all k = 1, : : : ,M ,

xk,0 2Argmaxxk,0

TXt=1

mtXj=1

log[p(zj(t) j xk,0)]wj,k(t):

(24)

C. EM in the Case of Maximum A Posteriori

Throughout the discussion, we only deal withthe ML. An easy extension can be introduced to takeadvantage of knowledge of prior probability relativeto the parameters. We show that a MAP derivation ofthe algorithm corresponds to the PMHT of Streit, et al.which results in Kalman filtering in the linear Gaussian

case [19]. Let us denote P(O) ¢=log[p(X ,¦)]. Supposeassignment probabilities don’t have prior probabilitiesas in [19], and the different models follow a Markovprocess5

P(O) ¢=log[p(X ,¦)]

= log

"p(X(0))

TYt=1

p(X(t) j X(t¡ 1))#

= log

"MYk=1

p(xk(0))MYk=1

TYt=1

p(xk(t) j xk(t¡ 1))#

=MXk=1

log[p(xk(0))]

+MXk=1

TXt=1

log[p(xk(t) j xk(t¡ 1))]: (25)

The MAP can be straightforwardly embedded into theEM framework. The E-step of the algorithm is changed

5For the sequel, only target models are considered in priorprobability. The model devoted to false alarms has been droppedsince no kinematic parameter has to be estimated. Of course, sucha model is taken into account in assignment probabilities (seeSection VI).



in [4]:M(O j O¤)

¢=Q(O j O¤)+P(O): (26)

Substituting (20) and (25) in (26) leads to

M(O j O¤) =MXk=1

TXt=1

"mtXj=1

wj,k(t)

#log[¼k(t)]

+MXk=1

TXt=1

mtXj=1

log[p(zj(t) j xk(t))]wj,k(t)

+MXk=1

log[p(xk(0))]

+MXk=1

TXt=1

log[p(xk(t) j xk(t¡ 1))]:

The maximization with respect to ¦ and to X is alwaysdecomposed in independent maximizations coupled withweightings wj,k(t). Assignment probabilities are updatedaccording to (22). The state vectors for next iteration aregiven by

(x0k ¢ ¢ ¢xTk ) 2

argmaxXk

8>>><>>>:TXt=1

mtXj=1

log[p(zj(t) j xk(t))]wj,k(t)

+log[p(xk(0))]+

TXt=1

log[p(xk(t) j xk(t¡ 1))]

9>>>=>>>; ,

for all k = 1, : : : ,M: (27)

An interesting case is the linear Gaussian Markovprocess described by (2). Instead of maximizing directly(27), consider the exponential of this expression [19]:

p(x0k )TYt=1

(p(xk(t) j xk(t¡ 1))

mtYj=1

p(zj(t) j xk(t))wj,k (t)),

k = 1, : : : ,Mand notice that

mtYj=1

p(zj(t) j xk(t))wi+1j,k(t)

/mtYj=1

N (zj(t) jHkxk(t), (wi+1j,k (t))¡1Rk)

/N (zk(t) jHkxk(t), (mt¼i+1k (t))¡1Rk)

where

zk(t) =1

mt¼i+1k (t)

mtXj=1

wi+1j,k (t)zj(t):

This maximization can be solved by using the Kalmanfiltering where the measurement is replaced with themeasurement centroid zk(t) with covariance matrix Rkdefined as

Rk =Rk

mt¼i+1k (t)

:

D. Conclusion

In conclusion, based upon an initialization of theparameters denoted O(0) = (¦ (0),X (0)), the new parametersat step (i+1) are updated iteratively according to thefollowing.

1) Update of ¦:

¼i+1k (t) =1mt

mtXj=1

wi+1j,k (t) where

wi+1j,k (t) =¼ikp(zj(t) j xik(t))p(zj(t) jOi(t))

for all t = 1, : : : ,T and k = 1, : : : ,M: (28)

2) Update of X :For the MAP:

(xi+1k (0) ¢ ¢ ¢xi+1k (T))

2 argmaxXk

(p(xk(0))

TYt=1

p(xk(t) j xk(t¡ 1))

£mtYj=1

p(zj(t) j xk(t))wi+1j,k(t)

)for all k = 1, : : : ,M:

For the ML:

xk,0 2Argmaxxk,0

TXt=1

mtXj=1

log[p(zj(t) j xk,0)]wj,k(t)

(29)

for all k = 1, : : : ,M: (30)

Maximizations with respect to X depend on theapplication and require generally the use of iterativeoptimization algorithms. In the next section,multitarget tracking is applied to passive SONAR. Thecorresponding maximization problems are detailed.

VI. MULTITARGET TRACKING IN PASSIVE SONAR

By considering measurement history composed ofCartesian positions for targets moving in horizontalplane, multitarget tracking is considerably simplifiedsince observability is ensured and measurementequation is a linear function of target states. On thecontrary, receiver maneuvers are required for ensuringobservability in the bearings-only tracking context.Furthermore, target state estimation is much moresensitive because optimizations in the M-step areno longer linear. In this section, we assume that theobservability conditions are fulfilled.

A. Parameter Estimation

In this section, targets move with constant velocityvectors. The parameter vector is thus denoted O = (X0,¦)where X0 indicates the M initial target states as in the



Fig. 1. Scenario in TMA.

E-step of the previous section:

O = (x1,0, : : : ,xM,0,¦(1), : : : ,¦(T)):

We focus on the update of Xi+10 based upon Oi and dropthe subscript on the target model since the maximizationwith respect to X0 is decomposed into M maximizations.

Denote the receiver state (Fig. 1) as

xobs(t) = (rxo,ryo,vxo,vyo)T

and the target state

xs(t) = (rxs,rys,vxs,vys)T:

The relative state vector is thus defined as

x(t) = xs(t)¡ xobs(t) = (rx,ry ,vx,vy)T:

The measurement equation (2) is replaced with

z = ¯(t)+ n(t) (31)

where

¯(t) = arctan

μrx(t)ry(t)

¶:

In (31), n(t) is a white Gaussian noise with variance¾2(t). Since targets are supposed to move at constantvelocity, a ML approach is used. The M-step of thealgorithm consists in computing (24). Using (31), thefunctional f(Z j x0) which has to be maximized for eachmodel is

f(Z j x0) =TXt=1

mtXj=1

log[p(zj(t) j x0)]wi+1j (t) (32)

=TXt=1

mtXj=1

¡ 12¾2(t)

(zj(t)¡¯(x0, t))2wi+1j (t) + ¢ ¢ ¢ :

(33)

In (33), ¯(x0, t) is the target azimuth at time t for initialtarget vector x0. Terms independent of x0 are droppedfrom (33). Taking the partial derivative with respect to x0yields the gradient vector:

rx0f =

TXt=1

1¾2(t)

@¯(x0, t)@x0

mtXj=1

(zj(t)¡¯(x0, t))wi+1j (t)

which can be expressed in a vectorial form as

rx0f = (R¡1x0 Ax0 )

TN¡1Bx0 (34)

with

Ax0=264cos¯(x0,1) ¡sin¯(x0,1) ±tcos¯(x0,1) ¡±tsin¯(x0,1)

......

......

cos¯(x0,T) ¡sin¯(x0,T) T±tcos¯(x0,T) ¡T±tsin¯(x0,T)

375

Bx0=

0BB@Pm1

j=1(zj(1)¡¯(x0,1))wi+1j (1)

...PmTj=1(zj (T)¡ ¯(x0,T))w

i+1j (T)

1CCAN = diag(¾2(t)), t = 1, : : : ,T

Rx0= diag(r(t)), t = 1, : : : ,T:

In (34), Rx0 is the matrix of the range sources atsampling time (±t is the sampling period) for inital targetstate x0 while N is the measurement variance matrix ifwe suppose that all measurements within a scan have thesame variance.

Notice that the gradient of the expression we needto maximize in M-step differs from the gradient of theclassical MLE in bearings-only TMA [12] only by themeasurement term Bx0 . In cluttered environment, it isreplaced by a mixing proportion of all the measurementswith their assignment probabilities that they originatefrom the target of interest given the previous estimation.

A solution to this maximization problem is obtainednumerically with a Gauss—Newton algorithm improvedby the Levenberg—Marquadt method [5]. This methoduses a trust region strategy. Since each M-step may face“weak” estimability, this method is required to avoidunrealistic estimates. Each iteration of this algorithmconsists in computing

xl+10 = xl0¡ [ATxl0R¡1xl0N¡1R¡1

xl0Axl

0+¹xl

0Id]¡1

£ATxl0R¡1xl0N¡1Bxl

0

where Id is the matrix identity and ¹xl0is the current

value used to perturb the Hessian r2x0f. Note that

in evaluating r2x0f, terms @2¯(x0, t)=@

2x0 have beendropped.

Summarizing the previous results, the PMHTalgorithm combines EM and Gauss—Newton iterations.Inside each EM iteration, the algorithm have to computeM maximizations which can be easily implemented onparallel computer since calculations of the weightingswi+1j (t) can be computed separately.

B. Numerical Results

Whatever the detection system (SONAR or RADAR),the data association step is a high level one whose inputsare the outputs of low level processing including signal



Fig. 2. Initialized trajectories and corresponding azimuths (case 1).

processing (filtering, discrete Fourier transform (DFT))and array processing (beamforming, predetection, etc.).In a narrowband passive SONAR, roughly speaking,the signal processing step is composed as follows:signals received from a sensor array are filtered, thensampled and analyzed (e.g., spectral analysis). A classicalbeamforming is performed to estimate instantaneoustarget parameters (bearings, etc.). Array processing alsoincludes a predetection step.

Our aim is now to define a realistic sonar simulation.So, the basic level of our simulations is the vectorformed from a single DFT bin from each sensor; thisvector is often called the snapshot vector. It is denoted X.

An assumption, common in the passive arrayprocessing context, asserts that this vector is zero-mean,Gaussian circular with covariance ¡ given by

¡ =MXi=1

¾iD¯iD¤¯i+ b Id

where * indicates transposition and conjugation.In the above formula, D¯i denotes the steering

vector of target i (M targets present in the surveillanceregion) and, in the case of a linear array,6 is given by theclassical formula:

D¯i¢=[1,exp(¡2i¼fs), : : : ,exp(¡2i¼(nc¡ 1)fs)]T

where fs stands for the spatial frequency (fs = cos¯i=2,intersensor distance d = ¸=2). The parameter ¾irepresents the power of target i while b is the noisepower.

In classical beamforming the spatial density of thefield impinging the array in the direction ¯ is estimatedby the value of the following quadratic form:

P(¯) =D¯¡D¤¯:

In this expression, ¡ is the estimated cross-spectralmatrix of the sensor outputs. If the averaged periodogramis considered, ¡ takes the following form:

¡ =1N

N¡1Xt=0

XtX¤t

6Linear array constituted of nc equispaced sensors.

where Xt is the tth vector of sensor DFT (N is thenumber of integrations).

1) Case 1 : Pd = 1 and Pfa= 0: For all thesimulations, the array processing parameters take thefollowing values: nc = 32, N = 125. In order to evaluatethe ability of the data-association algorithm for multiplesources, false alarms are not included at first. Extensionof measurement generation to cluttered environment isstraightforward.

Finally, to initialize the parameters of the algorithm,we consider that each model is equally probable and thatno strategy is applied for target state X00 . However, poorguess of the initial state may have dramatic effect on theestimated source states and assignment probabilities. Toavoid this drawback, we replace the direct maximization

of the likelihood functional L(O) ¢=log[p(Z j O)] by asequence of intermediate maximixations of functionalsfLi(O)gi=1¢¢¢N . In these functionals the measurementcovariance is increased to take into account moremeasurements in the evaluation of the weightingswi+1j,k (t). This idea of minimizing the influence of targetstates initiation have been previously derived in theprobabilistic data association (PDA) context [7]. Moreprecisely, this sequence is defined as follows: Li(O) =L(O,ai¾) where the faigi=1¢¢¢N is a decreasing sequencesuch that aN = 1. The state estimate obtained from Li(O)serves as the target state initiation for Li+1(O).

Fig. 2 presents a scenario for three targets withdifferent signal to noise ratios. The first targetis travelling from position (2000 m,25000 m)T

with speed (7 m/s,¡3 m/s)T, the second one from(11000 m,12000 m)T with speed (¡3 m/s,8 m/s)Tand the third one from (20000 m,10000 m)T withspeed (¡2 m/s,9 m/s)T. The signal-to-noise ratiosare assumed constant during the simulation, equal to(¡10 dB,¡15 dB,¡15 dB)T, respectively. Correspondingmeasurements are also shown in Fig. 2. Even if thetarget powers are quite similar, it is worth stressing theabsorption of weak targets by powerful ones, whichresults in a track coalescence in crossing areas. Initializedtarget trajectories and corresponding azimuths (values seeTable I) are displayed in Fig. 2.

The receiver trajectory is composed of three legsto ensure system observability. Its speed vector is(7 m/s,0 m/s)T for odd leg and (0 m/s,7 m/s)T for evenleg. The duration of the simulation is 1800s and thesampling period is ±t= 16 s. The results are presented



Fig. 3. Estimated trajectories and corresponding azimuths (case 1).

Fig. 4. Estimated probabilities f¼m(t)gt=1¢¢¢T,m=1¢¢¢M (case 1).

TABLE IInitiation of Algorithm X00

in Figs. 3 and 4. Despite a poor initiation, the algorithmconverges to the global solution. Although local solutionsor stationary points are possible, the use of the sequenceof functions defined above reduces their numbers.Despite its simplicity, this simulation demonstrates theability of the algorithm to resolve data-association indifficult crossing areas. Some interesting remarks canbe made by considering simultaneously Figs. 3 and 4.

REMARKS.1) For tracks well separated, the probability that a

measurement originates from each model is equallyprobable as it was expected and is verified up to t '11 mn, and from t' 18 mn to t ' 27 mn.2) From t' 10 mn to t ' 18 mn, model 2 and model

3 share the same measurements which are all assignedto the first one, thus assignment probability of model 2

is equal to 2/3, for model 3 to zero, and consequentlyfor model 1 to 1/3. The same remark can be made formodel 2 and model 1 in the other crossing area definedfor t ' 25 mn up to the end of the scenario. Note in theseareas, the instant when models separate and the changein their corresponding probabilities.3) Finally, as a conclusion of these remarks, we can

analyze the track coalescence effects with respect totarget powers. In the simulation, the most powerfultrack (model 2) is poorly estimated in comparison withweaker ones (model 1 and model 3). In other words,state estimate of the most powerful target is biased dueto the bias of bearing estimates in crossing area. Theprobabilities of model 1 and model 3 are very weak inthese areas, so their initial states are estimated basedupon unbiased measurements. The less the differencebetween target powers is, the more significant the bias is.Moreover, this phenomena is increased with the status of“weak” estimability.

2) Case 2 : Multitarget Tracking in the Presence ofFalse Detections: In the previous section, the target’sprobability of detection, Pd, was assumed equal tounity and no false alarms were generated. In thissection, the measurements that do not originate fromtargets are independently and uniformly distributed inthe measurement space, denoted U . The number offalse measurements in each scan followed a Poissondistribution with parameter ¸, where ¸ was equal to theexpected number of false alarms per unit volume. The



Fig. 5. Simulated measurements (case 2).

same measurement vector was used in this section witha target detection probability fixed to Pd = 0:8 for allthe targets during the batch. Although this simulationdoes not integrate the detection step, it gives a goodapproximation of reality. Fig. 5 depicts the simulatedmeasurements for the same scenario as in the previoussection.

Let us now describe the modifications of thealgorithm to include false alarms. Suppose the (M ¡ 1)first models describe target models and the Mth one isdevoted to false alarms. Since no kinematic parameterhas to be estimated for such a model, the M-step ofthe algorithm always consists in the same numberof optimizations. Moreover, for each target the samefunctional has to be maximized. It is straightforward toshow that the modification of the algorithm only liesin the evaluation of the assignment probabilities. Theconditional probability density of a measurement, giventhe previous parameter estimate and the model fromwhich it is originated, is given below

p(zj(t) j xk(t))

=

8><>:1

(2¼¾2)1=2exp

½¡ 12¾2

(zj(t)¡¯(xk0, t))2¾, 1· k <M

1U, k =M:

(35)

Thus, the a posteriori probabilities that a measurementoriginates from a target conditioned on the previousparemeter estimates is as follows:

wi+1j,k (t) =¼ikp(zj(t) j xik(t))p(zj(t) jOi(t))

wi+1j,k (t) =

8>>>>>>>>>>>><>>>>>>>>>>>>:

¼ik(t)PUk

exp

½¡ 12¾2

(zj(t)¡¯(xk0, t))2¾

¼iM(t)(2¼¾2)1=2

U+PM¡1

s=1

¼is(t)PUs

exp

½¡ 12¾2

(zj(t)¡¯(xs0, t))2¾ , 1· k <M

¼iM(t)(2¼¾2)1=2

U

¼iM(t)(2¼¾2)1=2

U+PM¡1

s=1

¼is(t)PUs

exp

½¡ 12¾2

(zj(t)¡¯(xs0, t))2¾ , k =M:

(36)

In (36), PUs is the normalization probability (i.e., theprobability that the measurement originates from thegiven model and lies in the surveillance region) defined

by

PUs =ZU

1(2¼¾2)1=2

exp

½¡ 12¾2

(z¡¯(xk0, t))2¾dz:

For the given scenario, initiation of the algorithmis presented in Fig. 6 and the estimated trajectories inFig. 7. This simulation demonstrates that the algorithm isvery good at solving the data association even in denselycluttered environment. Although the global solution isreached in this example, the algorithm may convergeto local solutions or stationary points of the likelihoodfunction. It is worth stressing that poor initializations canlead to poor data-association estimates, and thus, poortarget state estimates.

VII. EXTENSION TO MULTISENSOR DATAASSOCIATION

From previous sections, the data association problemhas focussed on temporal association of measurements.It is an easy task to extend the approach to spatial dataassociation of measurements from multiple sensors. Thepurpose is to study if we favour the solution to the dataassociation problem by providing informations from Ssensors. We show that the incomplete data frameworkextend easily to multisensor tracking. For that purposewe introduce new notations:

² Suppose we have S sensors.² Z = fZsgs=1:::S where Zs denote the set of

measurements received from sensor s.² X denote always the state vector of the M models.² ms,t is the number of measurements received by

sensor s at time t.² Ks is the assignment vector of measurements

received by sensor s.² ¦ denote assignment probabilities of the

measurements to the M models.We suppose that the set of measurements received

from different sensors are statistically independent. Asa consequence, we consider the assignment vector ofmeasurements Zs independent from the assignmentvector of Zs0 (s 6= s0). Thus, the assignment problembecomes a 2-dimensional data association problem

(temporal and spatial). Moreover, we consider acentralized fusion where the assignment probabilities areestimated based on all the measurements received by the



Fig. 6. Initialized trajectories and corresponding azimuths (case 2).

Fig. 7. Estimated trajectories and corresponding azimuths (case 2).

centralized system. From this viewpoint, the necessarylikelihoods are redefined so as to derive EM algorithm.The likelihood of the complete data expresses as:

p(Z,K j O) = p(Z1 : : :ZS ,K1 : : :KS j O)

=SYs=1

p(Zs j Ks,O)p(Ks j O) (37)

=SYs=1

TYt=1

ms,tYis=1

p(zis (t) j xkis (t))¼kis (t): (38)

Then, the likelihood of the incomplete data is obtainedby extending (16) to S sensors:

p(Z j O) =SYs=1

p(Zs j O)

=SYs=1

TYt=1

ms,tYis=1

p(zis (t) jO(t))

=SYs=1

TYt=1

ms,tYis=1

MXm=1

p(zis (t) j xm(t))¼m(t): (39)

Finally, the density of the missing data is deduced from(38) and (39):

p(K j Z,O) = p(Z,K j O)p(Z j O)

=SYs=1

TYt=1

ms,tYis=1

p(kis (t) j zis (t),O(t)) (40)

where

p(kis (t) j zis (t),O(t)) =p(zis (t) j xkis (t))¼kis (t)PM

m=1p(zis (t) j xm(t))¼m(t):

(41)

Thus, the calulation of Q(O j O¤) during E-step becomes:

Q(O j O¤) =XK

log[p(Z,K j O)]p(K j Z ,O¤)

=XK

(SXs=1

TXt=1

ms,tXis=1

log[p(zis (t) j xkis (t))¼kis (t)]

)

£

(SYs=1

TYt=1

ms,tYis=1

p(kis (t) j zis (t),O(t))

):

The calculation derived in appendix extendsstraightforward from the previous expressions sincethe decomposition required for this calculation remainsvalid. The simplified expression is therefore:

Q(O j O¤) =MXk=1

TXt=1

"SXs=1

ms,tXis=1

wis ,kis(t)

#log[¼k(t)]

+MXk=1

TXt=1

SXs=1

ms,tXis=1

log[p(zis (t) j xkis (t))]wis ,kis (t):

(42)

In (42), wis ,kis (t) is the weight of the measurement is fromsensor s and for the model kis , or more precisely thea posteriori probability to assign this measurement to



Fig. 8. Measurements and initialized trajectories for two sensors.

Fig. 9. Estimated trajectories and their corresponding azimuths for two sensors.

model kis :

wis ,kis(t) =

¼¤kisp(zis (t) j x

¤kis(t)

p(zis (t) jO¤(t)):

In the passive sonar context, if we suppose modelM is devoted to false-alarms, we show that the updatedprobability of model k, ¼k(t), is obtained by summing thea posteriori probabilities to assign all the measurementswithin a scan and from all the sensors to model k

¼i+1k (t) =1

(PS

s=1ms, t)

SXs=1

ms,tXis=1

wi+1is ,kis(t): (43)

In (43), wi+1is ,kis is calculated based on (36) where themeasurement equation is replaced by the measurementequation of sensor s. Moreover, the maximization withrespect to all the target state vectors is modified byadding the measurement equation for each sensor.We notice that the maximization of (42) can again bedecomposed in a maximization for each model. Sincesources are moving at constant velocity, the function tomaximize for a model k 2 f1 : : :M ¡ 1g defined by (33) ismodified for S sensors in:

f(Z j x0) =TXt=1

SXs=1

ms,tXis=1

log[p(zs,is (t) j x0)]wi+1is(t) (44)

=TXt=1

SXs=1

ms,tXis=1

¡ 12¾2s (t)

(zs,is (t)¡¯s(x0, t))2wi+1is (t)

+ ¢ ¢ ¢ : (45)

The gradient of (45) expressed in a vectorial form is

then:

rx0f =

SXs=1

(R¡1s,x0As,x0 )TN¡1

s Bs,x0 (46)

where the terms in (46) for a sensor s have the samemeaning as in (34).

We apply again a Gauss—Newton algorithm modifiedby the Levenberg—Marquadt method to solve themaximization problem. Each iteration consists incomputing:

xl+10 = xl0¡"

SXs=1

ATs,xl

0R¡1s,xl

0N¡1s R¡1

s,xl0As,xl

0+¹xl

0Id

#¡1

£SXs=1

ATs,xl

0R¡1s,xl

0N¡1s Bs,xl

0:

In summary, we have proved that the method extendseasily to multisensor problems. More precisely, we haveproposed a general framework to solve the difficultmultitarget multisensor tracking problem. However,the numerous simulations to which the algorithm wassubmited underscore that adding new sensors do notcontribute to the improvement of the solution of the dataassociation problem. The algorithm converges regularlyto some stationary points or local solutions preventingan accurate estimation of target parameters. We presenta simulation on Figs. 8 and 9 where the algorithmconverges to the global solution. This scenario was builtbased on the scenario of the previous section in the caseof one sensor. Here, the detection probability was equalto unity and 15 false alarms were generated in each scan.It brings to us the advantage of adding another sensorin terms of the quality of target parameter estimation assoon as data association is correctly estimated.



VIII. CONCLUSION

The multitarget tracking has been considered in thenatural framework of incomplete data problems. Basedon the approach of Streit, et al., we have developed amultitarget multisensor tracking method for the passivesonar context. More precisely, the formalism of Streit,et al. has been extended by using the general frameworkof mixture parameter estimation. The assignmentvector is not incorporated into the parameter vectorbut is interpreted as the missing data information.Thus, target state vector is estimated jointly withthe probabilities that a measurement comes from atarget. The use of the EM algorithm to compute thelikelihood function or the MAP, allowed us to derivetwo different algorithms which appear very appealingbecause the global maximization with respect to allthe target states is divided into M maximizations withrespect to each target state, each coupled thanks to theweightings wj(t). An extension of the temporal dataassociation to the spatial data association has beeninvestigated. This framework allows us to formulatethe multisensor multitarget tracking problem.Simulation results for realistic scenarios have beenconsidered. They demonstrate the good ability of thePMHT algorithm to perform multitarget tracking incrossing areas. Such simulations permit us to studysignal processing effects on multitarget trackingperformance in an integrated way. Some more workneed to be pursued in the evaluation of the trackingperformance of the algorithm and in the improvementof its robustness.

APPENDIX A

This Appendix is devoted to the calculation of theexpectation of the E-step in mixture densities. Extensionto multitarget tracking is straightforward. At the E-step,we need to evaluate:

Q(Á j Ái) =Xk

log[f(x j Á)]p(k j y,Ái)

=mX

k1=1

¢ ¢ ¢mX

kN=1

(NXj=1

log[¼kjpkj (yj j 'kj )])

£NYj0=1


p(yj0 j Ái):

This expression can be simplified by considering theN sums on one of the element of the expression interbraces:

Q(Á j Ái)

=NXj=1

mXk1=1

¢ ¢ ¢mX

kN=1

log[¼kj pkj (yj j 'kj )]

£NYj0=1


p(yj0 j Ái)

=NXj=1

8>><>>:mXkj=1

log[¼kj pkj (yj j 'kj )]¼ikj pkj (yj j '

ikj)

p(yj j Ái

£mX

k1=1

¢ ¢ ¢mX

kj¡1=1

mXkj+1=1

¢ ¢ ¢mX

kN=1

NYj0=1j0 6=j


p(yj0 j Ái)

9>>=>>; :(47)

The previous expression comes from a factorization ofthe terms in kj .

Using (7), the calculation of Bj defined as

Bj =mX

k1=1

¢ ¢ ¢mX

kj¡1=1

mXkj+1=1

¢ ¢ ¢mX

kN=1

NYj0=1j0 6=j

¼ikj0pkj0(yj0(yj0 j 'ikj0 )p(yj0 j Ái)

is an easy task since

Bj =mX

k1=1

: : :

mXkj¡1=1

mXkj+1=1

¢ ¢ ¢mX

kN=1

£p(k1, : : : ,kj¡1,kj+1, : : : ,kN j y,Ái) = 1 (48)

and then by substituting (48) in (47) and inverting sumsyields (9).

ACKNOWLEDGMENTS

The authors would like to thank R. L. Streit for allthe insightful comments that helped in enhancing thequality of this paper.

REFERENCES

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Herve Gauvrit graduated from the Ecole Superieure d’Electronique de l’Ouest, Angers,in 1992, and received his Diplome d’Etude Approfondie in signal processing from theUniversity of Rennes 1 in 1994.

He joined IRISA in 1994 and is currently in the Ph.D. program at the University ofRennes 1. His current research interests are in data association, estimation theory and itsapplications in multitarget multisensor tracking.

Claude Jauffret was born on March 29, 1957, in Ollioules, France. He receivedthe Diplome d’Etudes Approfondies in applied mathematics from the Saint CharlesUniversity, Marseille, France in 1981 and the Diplome D’ingenieur from the EcoleNationale Superieure d’Informatique et de Mathematiques Appliquees de Grenoble in1983, the title of “docteur de l’universite de Toulon” in 1993 and the “habilitation adiriger des recherches” in 1996 from the University of Toulon.

From Nov. 1983 to Sept. 1984, he worked on image processing used in passivesonar systems at the GERDSM, Six-Fours Les Plages, France. From 1984 to 1988, heworked on target motion analysis (TMA) problems at the GERDSM. A sabbatical yearat the University of Connecticut, Storrs, allowed him to work on tracking problems in acluttered environment in 1989. His research interests are mainly estimation and detectiontheory.

He has published papers about observability and estimation in nonlinear systems andTMA. Since September 1996, he is with the University of Toulon (France) where he is aProfessor.

J. P. Le Cadre (M’93) received the M.S. degree in mathematics in 1977, the Doctorat de3¡eme cycle in 1982 and the Doctorat d’Etat in 1987, both from INPG, Grenoble, France.

From 1980 to 1989, he worked at the GERDSM (Groupe d’Etudes et de Rechercheen Detection Sous-Marines), a laboratory of the DCN (Direction des ConstructionsNavales), mainly on array processing. Since 1989, he has been with IRISA/CNRS, wherehe is a director of research. His interests have now moved toward topics like systemanalysis, detection, data association and operations research.

He received (with O. Zugmeyer) the Eurasip Signal Processing best paper award in1993. He is a member of various societies of IEEE and of SIAM.

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