Fast multiple target tracking using particle filtersMark R. Morelande∗ and Darko Musicki∗

Melbourne Systems LaboratoryDepartment of Electrical and Electronic Engineering

The University of Melbourne, Australiaemail: m.morelande@ee.unimelb.edu.au, d.musicki@ee.unimelb.edu.au

Abstract— A fast algorithm for accurate initiation andtracking of multiple targets is developed. The basis ofthe approach is the linear multi-target (LM) method inwhich exact joint data association for multiple targets isapproximated by single target data association for eachtarget with clutter density modified to include contributionsfrom neighbouring targets. This enables multiple targettracking to be performed with the computational expense ofsingle target tracking. Previously the LM method has beencombined with tracking using a Gaussian approximation. Inthis paper accurate tracking is achieved through the use of avariant of the auxiliary particle filter. Simulation results showthat the use of the particle filter is particularly advantageousin demanding scenarios where targets are in close proximityfor long periods.

I. INTRODUCTION

The main difficulty in multiple target tracking is posedby measurement origin uncertainty. This arises becauseeach measurement can be due to one of several targetsor to some non-target related phenomena. Measurementsfrom non-target related phenomena are referred to asclutter. The optimal way of resolving measurement originuncertainty is to enumerate and evaluate all possibleassociations between the measurements and the targets. Inan environment in which the number of targets is unknownit is also necessary to enumerate hypotheses proposingthe existence of new targets. Exhaustive enumeration ofall possible association hypotheses is an impossible taskand sophisticated hypothesis management is required [2],[15]. Algorithms which use this approach are collectivelyreferred to as multiple hypothesis trackers (MHTs).

The difficulty of developing hypothesis managementtechniques which reduce computation while maintainingperformance led to the development of the joint integratedprobabilistic data association filter (JIPDAF) [12]. TheJIPDAF uses as its basis the joint probabilistic dataassociation filter (JPDAF) [9]. The JPDAF tracks a fixednumber of targets by enumerating all measurement-targetassociations at each scan and then combining them intoa single component. This can be regarded as an extremeform of hypothesis management. The JIPDAF extends theJPDAF through the calculation of an “existence probabil-ity” which enables target tracks to be initiated, confirmedand terminated depending on the received measurements.

The requirement for joint association of several mea-surements with several targets limits the scenarios in

which the MHT and JIPDAF can be applied with rea-sonable computational expense. In particular, the numberof targets in a given area cannot be too large since thenumber of association hypotheses increases exponentiallywith the number of targets. There has therefore been muchinterest in developing methods for performing joint dataassociation in such a way that this exponential increase incomputational expense is avoided with a minimal loss ofaccuracy [8], [16], [17], [18].

More recently, a technique which permits approximatejoint data association with a computational expense whichis linear in the number of targets was proposed in [13].This technique will be referred to as linear multi-target(LM) tracking. The key idea in LM tracking is thateach target performs single target data association withthe neighbouring targets treated as clutter. This takesinto account the presence of neighbouring targets withoutincurring exponentially increasing computational expense.In fact, LM tracking for multiple targets is only slightlymore computationally expensive than performing singletarget data association for each target. The algorithmsof [8], [17] also have a computational expense which islinear in the number of targets but perform poorly whena large number of targets are in close proximity [16]. Therobustness of LM tracking to increases in the number oftargets is demonstrated in [11].

As with any data association technique, optimal targetstate estimation using the LM method, in the Bayesiansense, requires a computational expense which increasesexponentially with time. This prevents exact computationof the posterior distribution of the target state. In [13],probabilistic data association (PDA), in which the poste-rior density is approximated by a Gaussian by combiningthe association hypotheses at each time step [1], was used.The combination of the LM method with IPDA is referredto as the LMIPDAF. PDA is a reasonable approximationbut tends to break down in more demanding scenariosinvolving large numbers of targets and/or clutter measure-ments. In such cases the use of a numerical techniquewhich permits arbitrarily accurate approximation of theposterior distribution, albeit at additional computationalexpense, is desirable. This approach is pursued here witha particle filter (PF) used to approximate the posteriordistribution of the target state. The resulting algorithm isreferred to as the LM integrated PF (LMIPF).

Proceedings of the44th IEEE Conference on Decision and Control, andthe European Control Conference 2005Seville, Spain, December 12-15, 2005

MoA16.1

0-7803-9568-9/05/$20.00 ©2005 IEEE 530

Essentially, PFs use Monte Carlo simulation to approx-imate the integrals which arise in the evaluation of theposterior distribution [6]. This results in a representationof the posterior distribution comprised of a number ofweighted random samples, or particles. The advantagesof PFs, in comparison to other numerical techniques [3],lie in the ease with which they can be implemented, thegenerality with which they can be applied and their ability,when properly designed, to reduce the effects of the curseof dimensionality [5]. Convergence results for PFs undera wide range of conditions can be found in [4].

PFs are commonly implemented using the sequentialimportance sampling (SIS) framework which involves par-ticle proposal, weight update and possibly resampling ateach time step. Depending on the manner in which the PFis implemented data association is performed in the weightupdate, as in the bootstrap filter [10], or is part of both theparticle proposal and weight update, as in a measurement-directed proposal [7]. A measurement-directed proposalprovides superior performance for a given sample sizeand will be the approach used here. In particular, avariant of the auxiliary PF [14] is proposed. LM dataassociation, which permits particles to be proposed andupdated separately for each target using single target dataassociation, greatly reduces the computational expense ofthe particle proposal and weight update steps. Simulationswill demonstrate that this decrease in computational ex-pense is achieved while maintaining good performance.

The paper is organised as follows. Section II definesthe notation. Section III contains a brief review of theLM method. The particle filtering algorithm is describedin Section IV. Section V contains the simulation resultsand conclusions are drawn in Section VI.

II. NOTATION

Measurements are made at times t1, t2, . . .. At eachtime step the LM algorithm maintains a number of tenta-tive tracks. Let rk denote the number of tentative tracks attime tk. The relevant information concerning the ith trackis contained in the dynamic state xk,i and the existencestate. The dynamic state contains kinematic informationsuch as target position and velocity and manoeuvringmode. The binary existence state permits track formationand deletion. The event that the ith track is following atarget, i.e., the ith track exists, is denoted as χk,i withχk,i denoting the converse.

Assume that the ith track is initiated at time k0. Thenthe existence state of the ith track evolves as

P(χk,i|χk−1,i) = η, (1)

P(χk,i|χk−1,i) = 1, k = k0 + 1, k0 + 2, . . . , (2)

with P(χk0,i) = ρ0. Conditional upon target existence,the dynamic state xk,i evolves according to

xk,i|xk−1,i ∼ N(fk(xk−1,i), Qk), (3)

for k = k0 + 1, k0 + 2, . . . , where N(µ, Σ) is theGaussian distribution with mean µ and covariance matrix

Σ. If necessary, the transition probability (1) and transitiondensity can differ between targets. At time k0, xk0,i ∼τk0,i.

At time tk, mk measurements are received and collectedinto yk = {yk,1, . . . , yk,mk

}. The ith target is associatedwith a set of mk,i validated measurements which areobtained by performing gating in the usual manner. LetGk,i ⊆ {1, . . . , mk} denote the set of indices of themeasurements validated by the ith target. Then, θj,i, i =1, . . . , rk, j ∈ Gk,i is the event that the jth measurementis due to the ith target and θ0,i denotes the event that theith target has not been detected. Considering the ith targetin isolation we have, for j, l ∈ Gk,i,

yk,l|xk,i, θj,i ∼{

N(h(xk,i), Rk), l = j,ck, otherwise,

where ck is the assumed probability density function forclutter measurements. Under θ0,i, yk,l ∼ ck for l ∈ Gk,i.Note that θj,i, j ∈ {0}∪Gk,i are the complete collectionof single target association hypotheses for the ith target.Thus if the jth measurement is not due to the ith targetit is a clutter measurement. In multi-target tracking itis also necessary to consider the possibility that the jthmeasurement is due to some other target. The optimalapproach does this by enumerating additional associationhypotheses. In order to reduce computational expensethe LM method does it by modifying the clutter densityaccording to the proximity of neighbouring targets. Thisis described in Section III.

Targets are detected with probability PD at each timestep. If a target is detected, the probability that thetarget measurement falls in the validation gate of thattarget is PG. When the notation p is used to denoteprobability density functions the particular density underconsideration will be clear from the arguments.

The goal of the tracking algorithm is to recursivelycompute the posterior density of the dynamic state, de-noted as πk|k,i at time tk for the ith target, and theposterior probability of track existence for each tentativetrack. Here the LM method is used in conjunction with aparticle filter to compute these quantities. The proceduresare described in the following two sections.

III. REVIEW OF LINEAR MULTI-TARGET TRACKING

The key idea in LM tracking is that when updatingone track, detections of targets followed by other tracksare unwanted measurements or clutter. Clutter measure-ment density is modulated to account for the additionalsource(s) of clutter, after which single target tracking isperformed.

Denote with yk the set of measurement up to andincluding time tk:

yk =k⋃

t=1

yt.

Letνk,i(j) =

∫p(yk,j |x)πk|k−1,i(x) dx (4)

531

denote the a priori target measurement density for theith track evaluated at the jth measurement. Denote theclutter measurement density at the jth measurement asρk(j) = ck(yk,j).

The first step of the LM approach is to calculate apriori data association probabilities for each track, underthe assumption that the selected measurements may haveoriginated from the target or from clutter only. Under thisassumption, the a priori probability that the jth measure-ment is the detection of the ith target is approximated by,for j ∈ Gk,i,

P ij ≈ PDPGP

(χk,i|yk−1

) νk,i(j)ρk(j)∑

l∈Gk,i

νk,i(l)ρk(l)

. (5)

The a priori clutter measurement density of the jth mea-surement when updating the ith track is, after includingthe contributions of neighbouring targets, given by

ρk,i(j) = ρk(j) +rk∑

d=1d �=i

νk,d(j)P d

j

1 − P dj

. (6)

The LM method involves using ρk,i(j) rather than ρk(j)when computing data association probabilities for the ithtrack and otherwise ignoring neighbouring tracks.

Automatic track formation is performed by initialisingnew tracks at each time step using two-point initiation.These new tracks are given a prior target state density,based on the two measurements used for initialisation,assigned an initial existence probability and added to theset of tentative tracks. Tracking of the existing tentativetracks is performed using the algorithm of Section IV withthe existence probability updated using (19). Tentativetracks are classified as confirmed, i.e., they are deemedto belong to a target, or are terminated on the basis of theexistence probability. Confirmation occurs when the exis-tence probability exceeds a threshold Pc while terminationoccurs if the existence probability falls below a probabilityPt. This simple track management is augmented withprocedures for merging tracks which are very close forlong periods of time. Details can be found in [13].

IV. PARTICLE FILTERING USING THE LM METHOD

In a PF, the posterior density of the ith targetstate at time tk−1 is represented by a set of particlesx1

k−1,i, . . . , xnk−1,i and weights w1

k−1,i, . . . , wnk−1,i where

n is the sample size. Upon receipt of measurements at timetk, these particles and weights are modified to produce anapproximation to the posterior density. In SIS the new par-ticles and weights are produced by importance samplingwhich involves drawing samples from a proposal or im-portance density and applying a multiplicative adjustmentto the weights. To prevent sample degeneracy, in whichthe weights become concentrated in only a few particles,resampling should be performed regularly. Resampling

involves selecting particles according to the weights sothat particles with small weights will be removed whilethose with large weights will be selected many times.

A. Target state posterior density approximation

The best importance density to use, in the sense that itminimises the variance of the particle weights conditionalupon the measurement history to time tk and the particletrajectory to time tk−1, is the optimal importance density(OID) [7]. However, the OID can be used only if themeasurement equation is linear and Gaussian. In thefollowing we use a variant of the auxiliary PF (APF) [14]to provide a natural extension of the OID for nonlinearmeasurement equations. The proposed method reduces tothe OID when the measurement equation is linear.

The PF approximation to the posterior density of theith target state at time tk−1 can be written as

πk−1|k−1,i(x) =n∑

t=1

wtk−1,iδ(x − xt

k−1,i). (7)

Using Bayes’ rule leads to the following approximationto the posterior density at time tk:

πk|k,i(x) ∝ p(yk|x)n∑

t=1

wtk−1,ip(x|xt

k−1,i). (8)

Equivalently,

πk|k,i(x, t) ∝ p(yk|x)p(x|xtk−1,i)w

tk−1,i, (9)

where t ∈ {1, . . . , n}, an index on the mixture in (8),is referred to as an auxiliary variable. It is proposed toobtain samples from (9) using importance sampling withan importance density of the form

q(x, t|yk) = γt(yk)g(x|xtk−1,i, yk), (10)

where∑n

t=1 γt(yk) = 1. A draw from (10) is made byselecting t = l with probability γl(yk) and then drawingxt

k,i from g(·|xtk−1,i, yk). The probabilities γt(yk), t =

1, . . . , n will be referred to as the first-stage weights. In[14], it is suggested to use first-stage weights of the form

γt(yk) ∝ wtk−1,ip(yk|µt

k), (11)

where µtk is some value which characterises the transition

density p(xk,i|xtk−1,i). Since the best approach, if possi-

ble, is to draw samples from the OID, it is of interestto examine the relationship between the use of (10) with(11) and the use of the OID. In the OID updated weightsare proportional to wt

k−1,ip(yk|xtk−1,i). The first-stage

weights used in the APF can be interpreted as an attemptto approximate this weight update. Thus the APF withfirst-stage weights as in (11) approximates the predictivelikelihood at measurement y as follows:

p(y|xk−1) =∫

p(y|xk, xk−1)p(xk|xk−1) dxk (12)

≈∫

p(y|xk, xk−1)δ(xk − µ) dµ = p(y|µ). (13)

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Thus, if the APF is regarded as an approximation tothe OID, implicit in the use of the weights (11) is anapproximation of the transition density by a delta functioncentred at some characteristic value of the state. This isa crude approximation which will, in general, result in asampling procedure substantially different from the OID.

Since the form of the likelihood is the barrier preventingexact computation of (12), the approach taken here is toreplace the likelihood by an approximate quantity whichallows analytic computation of the integral (12). This is astrategy used in nonlinear filtering approximations suchas the extended and unscented Kalman filters. In thispaper the likelihood is approximated by linearising themeasurement equation about the predicted target state, asin the extended Kalman filter. This requires computationof the Jacobian, denoted as H = (∇xh(x)′|x=f(xk−1))

′

with ′ denoting matrix transpose. Under the linearisedapproximation to the measurement equation, the first-stageweights are

γt(yk) = wtk−1,il(yk|xt

k−1,i)

/n∑

d=1

{wd

k−1,il(yk|xdk−1,i)

},

(14)where

l(yk|xtk−1,i) = 1 − PDPG

⎧⎨⎩1 −

∑j∈Gk,i

l(yk,j |xtk−1,i)

ρk,i(j)

⎫⎬⎭ ,

with ρk,i(j) given in (6) and

l(yk,j |xk−1,i) = N(yk,j ; h(fk(xk−1,i)), HQkH ′ + Rk).

In the remainder of the paper the notation l will be used inplace of p to denote pdfs derived using linearisation of themeasurement equation. In keeping with the desire for thesampling procedure to be as close to the OID samplingprocedure as possible, the density g used to propose thoseparticles selected by the first-stage weights is a linearisedapproximation to the OID:

g(x|xtk−1,i, y

k) =l(yk|x, xt

k−1,i)p(x|xtk−1,i)

l(yk|xtk−1,i)

=∑

j∈{0}∪Gk,i

αtk,i(j)N(x; µt

k−1,i(j), Σtk−1,i(j)),

(15)

where

αtk,i(j) =

⎧⎪⎪⎨⎪⎪⎩

1 − PDPG

l(yk|xtk−1,i)

, j = 0,

PDPGl(yk,j |xtk−1,i)

ρk,i(j)l(yk|xtk−1,i)

, j ∈ Gk,i,

µtk,i(j) =

⎧⎪⎪⎨⎪⎪⎩

f(xtk−1,i),

j = 0,f(xt

k−1,i) + Ktk,i{yk,j − f(xt

k−1,i)},j ∈ Gk,i,

Σtk,i(j) =

{Qk, j = 0,Qk − Kt

k,iHtk,iQk, j ∈ Gk,i,

withKt

k,i = QkHt′k,i(H

tk,iQkHt′

k,i + Rk)−1. (16)

While the Jacobian matrix H tk,i can be different for

each particle, for instance H tk,i = (∇xh(x)′|x=f(xt

k−1,i))′,

it is significantly simpler computationally to use the samematrix for all particles. Here we use, for t = 1, . . . , n,Ht

k,i = Hk,i = (∇xh(x)′|x=f(xk−1|k−1,i))′, where

xk−1|k−1,i =n∑

t=1

wtk−1,ix

tk−1,i. (17)

Thus the Jacobian, along with the gain matrix K tk,i, need

be computed just once for each target at each time step.The final step is to assign weights to the proposed

particles. The weight for the tth particle of the ith trackcan be found as

wtk,i ∝

πk|k,i(xtk,i, t)

q(xtk,i, t|yk)

=p(yk|xt

k,i)l(yk|xt

k,i, xtk−1,i)

. (18)

The ratio (18) accounts for the approximations involvedin linearising the measurement equation. A summary ofthe procedure for producing a particle approximation tothe posterior distribution of the ith target state at time tk

from a corresponding approximation at time tk−1 is givenin Table I.

TABLE I

A RECURSION OF THE PARTICLE FILTER

1) Compute the first-stage weights γ1(yk), . . . , γn(yk) using(14).

2) Select particle indices j1, . . . , jn such that jt = s withprobability γs(yk).

3) Draw target states xtk,i ∼ g(·|xjt

k−1,i, yk), t = 1, . . . , n withg given in (15).

4) Compute the updated weights w1k, . . . , wn

k using (18).

For a linear measurement equation, the linearised andtrue likelihoods are identical and the weights are even, i.e.,wt

k,i = 1/n. If the measurement equation is nonlinear, thevariability of the weights will depend on the accuracy ofthe linerisation. In the scenario of Section V the accuracyof the linearised measurement equation is sufficient toproduce almost equal weights for each particle. This isdesirable since highly variable weights reduce accuracy.

Although the use of linearisation to approximate theOID is not new, the procedure suggested here is subtlydifferent from existing methods. The difference betweenthe proposed technique and the APF as proposed in [14]has been described above. Linearisation of the measure-ment equation was also suggested in [7] although not in anauxiliary variable framework. An important aspect of theuse of auxiliary variables is that particles are selected forresampling, using the first-stage weights, prior to drawingnew samples. This ensures a diverse particle set, andimproved performance, since all particles will have uniquevalues. In [7] resampling is performed after drawing newparticles. This causes duplication of particle values and aloss of diversity.

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B. Computation of the existence probability

The posterior probability of target existence for the ithtrack is calculated as [11]

P(χk,i|yk

)=

(1 − δk,i)P(χk,i|yk−1

)1 − δk,iP (χk,i|yk−1)

, (19)

where

δk,i = PDPG

⎛⎝1 −

∑j∈Gk,i

νk,i(j)ρk,i(j)

⎞⎠ . (20)

The quantities ρk,i(j) are computed as in (6) while theprior probability of target existence is computed using (1)and (2). This leaves the quantities νk,i(j), j ∈ Gk,i. Recallthat νk,i(j) is the prior target measurement density for theith target computed at the jth measurement. This can beexpanded as

νk,i(j) =∫

p(yk,j |xk,i)πk|k−1,i(xk,i) dxk,i. (21)

An approximation to the density πk|k−1,i can beobtained by stochastic prediction of the particlesx1

k−1,i, . . . , xnk−1,i. This results in the approximation

πk|k−1(xk,i) =n∑

t=1

wtk−1,iδ(xk,i − xt

k,i), (22)

where xtk,i ∼ N(f(xt

k−1,i), Qk), t = 1, . . . , n. Replacingπk|k−1,i in (21) with the approximation (22) gives

νk,i(j) =n∑

t=1

wtk−1,ip(yk,j |xt

k,i). (23)

The quantities νk,i(j), j ∈ Gk,i are computed using(23) and used in place of the exact values in (20) toapproximate δk,i. This approximation is then substitutedinto (19) to obtain an approximation to the posteriorprobability of target existence.

V. SIMULATION RESULTS

In this section the performance of the LMIPF is ex-amined using Monte Carlo simulations. The previouslyproposed LMIPDAF is taken as a benchmark. It has beendemonstrated in [13] that the LMIPDAF is capable ofslightly better performance than the JIPDAF, which usesexact joint association for neighbouring targets. The supe-riority of the LMIPDAF over the JIPDAF arises becausethe computational efficiency of the LMIPDAF allows it toperform approximate multiple target data association onall tentative tracks while the JIPDAF can perform multipletarget data association only on confirmed tracks.

Measurements are collected at equi-spaced time in-stants, tk = tk−1 + T , k = 1, 2, . . . where T is thesampling period. Targets move in two dimensions witha constant velocity. The ith target state is comprisedof position and velocity in each direction and evolvesaccording to xk,i = Fxk−1,i, k = 1, 2, . . . with

F = I2 ⊗(

1 T0 1

),

where ⊗ is the Kronecker product and Im is the m × midentity matrix. Note that target trajectories are generatedwithout process noise but the filters assume process noiseso that target states are assumed to evolve according toxk,i|xk−1,i ∼ N(Fxi,k−1, Q) where

Q = I2 ⊗ 1/10(

T 3/3 T 2/2T 2/2 T

).

Target position is measured in polar coordinates usinga sensor located the origin of the reference frame. Themeasurement noise covariance matrix is

Rk = diag((π/180)2, 16).

Each target is detected with probability PD = 0.9 and thegating probability is set to PG = 0.99 in order to minimisethe number of validated measurements.

A scenario with five crossing target trajectories isconsidered. Targets move with a speed of 15 m/s andare tracked for 70 time steps with a sampling period ofT = 1s. The trajectories of all targets meet at the 40th timestep. The clutter is Poisson distributed with density 0.02pts/(scan·m·rad) throughout the surveillance region. Theangle between the neighbouring trajectories is used to varythe severity of the scenario. A small angular separationincreases the severity of scenario since it increases theamount of time that the targets share measurements.

Performance is measured by the mean number of con-firmed true tracks for a given number of confirmed falsetracks. A true track is one which has been initiated fromtarget measurements or has a state estimate sufficientlyclose to the true target state. The mean number of con-firmed true tracks is estimated over 1000 realisations ofthe scenario described above for the LMIPDAF and theLMIPF using 200, 500 and 1000 particles. The confir-mation and termination probabilities are such that thefalse track confirmation rate is one per 50 time steps.The results are given in Figures 1 and 2 for angularseparations of 10 degrees and 5 degrees, respectively. Bothalgorithms perform well with angular separations of 10degrees although it is clear that a sample size of at least500 particles is required for the LMIPF. The followingcomments regarding the performance of the LMIPF applyonly for these larger sample sizes. Compared to theLMIPF, the LMIPDAF is quicker to confirm tracks buthas a greater tendency to lose tracks when the targets arein close proximity. Evidence of this latter characteristic isthe perceptibly larger decrease in the number of confirmedtrue tracks before and after the 40th time step for theLMIPDAF compared to the LMIPF. The results for anangular separation of 5 degrees more clearly demonstratethe superiority of the LMIPF over the LMIPDAF fortracking in dense target environments. The mean numberof true tracks confirmed by the LMIPF is significantlylarger across the entire observation interval and the dip inperformance immediately after the target trajectories crossis less marked.

534

0 10 20 30 40 50 60 700

1

2

3

4

5

Time

Num

ber

of tr

ue tr

acks

con

firm

ed

Fig. 1. Mean number of confirmed true tracks plotted against time forthe LMIPDAF (dotted) and the LMIPF with 200 particles (dash-dot),500 particles (dashed) and 1000 particles (solid). The angular separationbetween neighbouring target trajectories is 10 degrees.

0 10 20 30 40 50 60 700

1

2

3

4

5

Time

Num

ber

of tr

ue tr

acks

con

firm

ed

Fig. 2. Mean number of confirmed true tracks plotted against time forthe LMIPDAF (dotted) and the LMIPF with 200 particles (dash-dot),500 particles (dashed) and 1000 particles (solid). The angular separationbetween neighbouring target trajectories is 5 degrees.

Computational expense is an important considerationwhen using a numerical method such as a particle filter.The results of Figures 1 and 2 indicate that significantimprovement in performance is obtained by increasingthe sample size of the LMIPF from 200 to 500 particlesbut only marginal improvement is obtained by a furtherincrease to 1000 particles. This implies that, for the sce-nario under consideration, a sample size of 500 particlesprovides the best trade-off between computational expenseand performance. With this relatively small sample sizethe computational expense of the LMIPF is about fourtimes that of the LMIPDAF. Even with this four-foldincrease in computational expense over the LMIPDAF,the LMIPF is remarkably computationally efficient com-pared to algorithms which perform exact multi-target dataassociation such as the MHT and JPDAF.

VI. CONCLUSIONS

An algorithm for automatic track formation and main-tenance of multiple targets was proposed. The algorithm

combines the linear multi-target method, so-called becauseit’s computational expense is linear in the number oftargets and measurements, with particle filtering to arriveat a fast accurate solution. The performance of the pro-posed algorithm was examined for a five-target scenariowith positions measured in polar coordinates. Simulationresults showed that, compared to an existing algorithmwhich employs a Gaussian approximation, the proposedalgorithm performs particularly well in demanding scenar-ios where targets are in close proximity for a long periodof time. Importantly, this performance is achieved with areasonably small sample size. Thus, combining the linearmulti-target method with particle filtering permits reliablemultiple target track formation and maintenance in densetarget environments with modest computational expense.

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