Multitarget Initiation, Tracking and Termination

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Multitarget Initiation, Tracking and Termination Using Bayesian Monte Carlo Methods WILLIAM NG *, J ACK LI , SIMON G ODSILL

AND

SZE K IM PANG

Department of Engineering, University of Cambridge, CB2 IPZ Cambridge, UK *Corresponding author: [email protected] In this paper, we present an online approach for joint initiation/termination and tracking for multiple targets with multiple sensors using sequential Monte Carlo (SMC) methods. There are several main contributions in the paper. The first contribution is the extension of the deterministic initiation and termination method proposed by the authors’ previous publications to a full SMC context in which track initiation/termination are executed with sampling methods. In effect, the dimensions of the particles are variable. In addition, we also integrate a Markov random field (MRF) motion model with the framework to enable efficient and accurate tracking for interacting targets and to avoid potential track coalescence problems. With the employment of multiple sensors, a centralized tracking strategy is adopted, where the observations from all active sensors are fused together for target initiation/termination and tracking and a set of global tracks is maintained. Intra- and inter-sensor clusters are constructed, comprised of closely spaced observations either in time for single sensors or from distinct sensors at a single time, that can increase the reliability when proposing new tracks for initiation. Computer simulations demonstrate that the proposed approach is robust in joint initiation/termination and tracking of multiple manoeuvring targets even when the environment is hostile with high-clutter rates and low target detection probabilities. The integration of the MRF framework into the proposed methods improves robustness in handling close target interactions when the observation noise is high. Keywords: multitarget tracking; multiple sensors; data and track fusion; Markov random field; particle filter; sequential Monte Carlo; clustering algorithm Received 22 May 2007; revised 22 May 2007

1. INTRODUCTION Multitarget tracking (MTT) [1–3] is an important element of surveillance and monitoring systems that require the determination of the number as well as the dynamics of targets. Applications include radar and sonar-based tracking of objects for navigation and air traffic control. In practice, a tracking system, which relies on a single sensor for target detection and tracking, is vulnerable to errors and noise problems. The shortcomings single-sensor systems suffer can be mitigated to a large extent when additional sensors are deployed. For instance, when multiple sensors are distributed at widely separated locations, the system is capable of providing a clearer and more reliable view of activities within the surveillance region than a single-sensor system. As a result, more accurate and reliable target detection and tracking can be achieved from multiplesensor (multisensor) systems. More detailed discussions on the

features and the types of multiplesensor tracking systems can be found in [4–6]. In reality, multisensor tracking applications face some significant challenges. First, the number of targets is unknown and must automatically be determined for successful operation. Second, the state-space models are often nonlinear and nonGaussian so that no closed-form analytic solution can be obtained. As a result, methods relying on the linear and Gaussian assumptions are susceptible to failure. Third, a real-world target often moves in straight lines with constant velocity and occasionally manoeuvres abruptly during its motion. A single dynamical model for modelling the target motions is not typically enough to represent this behaviour. To address this issue, interactive multiple models (IMMs) are proposed [7–10], where each model, representing a different type of target

THE COMPUTER JOURNAL, Vol. 50 No. 6, 2007

MULTITARGET INITIATION, TRACKING dynamics, is run in parallel, and the output of the algorithm is a weighted sum of the individual model filters. As an alternative, variable rate models attempt to model random manoeuvres of different types as piecewise deterministic functions with unknown parameters [11]. Although these approaches are intuitively reasonable, they perhaps unnecessarily complicate the model and increase the computational complexity. Fourthly, in most tracking algorithms, explicit data association between the active targets and the observations is needed prior to estimating the target estimates, since the observers usually yield unlabelled observations. Unfortunately, it is well known that even a moderate number of active targets and observations require intensive and complex computations to perform the data association. Finally, the tracking problem is further compounded when multiple sensors are deployed, because one needs to resolve the increased costs and complexity of data fusion and transformation as well as communications among the sensors. Attention has recently focused on the role of simulation-based approaches, including sequential Monte Carlo (SMC) methods, in solving the MTT problem, as these methods are able to perform well for nonlinear and non-Gaussian data models. Also known as Particle Filtering [12–15], SMC methods are based on sequential importance sampling to generate a set of samples, or particles, and their associated importance weights, which are then propagated through time to give predictions of the target posterior distribution function at future time steps. Various particle filtering strategies for dealing with an unknown and variable number of targets have been developed. Modelling the unknown number of targets as a random set, the Probability Hypothesis Density (PHD) filter [16 – 18] can jointly estimate the number of targets and their positions. Nevertheless, estimates of the target states are generally less accurate than those obtained by more conventional techniques, since target identity is not preserved. Furthermore, these methods suffer from performance degradation when the environment is characterized by higher clutter rate and low target detection probability [19, 20]. Other SMC-based alternatives are proposed in [20, 21], which jointly estimate a track existence variable to handle track initiation and termination and the multitarget state. Other methods based on particle filters for joint multiple target detection and tracking include [22 – 24]. A Jump Markov System [22], a very general framework for multiple target detection and tracking and sensor management, is proposed, but details on how new targets may be proposed are not given. In [23], while the authors propose to conduct hypothesis testing for target detection, they also acknowledge that the proposed methods do not perform automated detection and have not provided a complete solution to this problem. In [24], a joint multitarget probability density is proposed in which a single entity simultaneously captures the uncertainty about target number, target state and target identification. This proposed method, based on a track-before-detect strategy, divides the surveillance region into a number of cells

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and proposes new track initiation when activities are detected in these cells. Whereas the above SMC-based approaches have not explicitly dealt with the model-switching problem, other SMC-based IMM methods are proposed in [25 –28], but only the method in [26] performs automatic target detection. While providing new approaches to deal with data association problems, the method in [26], however, relies on the assumption that nearly linear models are available such that the weighted posterior mean and covariance are propagated over time to track the targets. In this paper, we propose a solution to the problem of joint initiation/termination and tracking for manoeuvrable targets using multiple sensors in the SMC framework. We extend the SMC methods in [29 – 32] with a single sensor to multiple sensors for automatic initiation/termination and tracking for multiple targets. We adopt a central-level tracking strategy [6] to collect and fuse all observations from the different sensors and perform a centralized target initiation/ termination and maintenance. The proposed strategy for target initiation/ termination uses a stochastic sampling method to propose either a birth move for track initiation, a death move for track termination or an update move for track maintenance for every particle, according to the persistence of the activities being detected. As a result, the particles have time-varying and different dimensions. Introducing inter-sensor measurement clusters which are composed of measurements from all active sensors at a given time, we can raise the confidence level at which detected activities originate from targets. It follows that the likelihood of initiating tracks for spurious objects is significantly reduced. With an estimated number of active targets in every particle, the proposed algorithm utilizes a data-dependent importance sampling method [29–32] to generate target state particles using latest observations grouped in intra-sensor measurement clusters. Unlike an inter-sensor cluster, every intra-sensor cluster comprises measurements from the same sensor that are grouped together according to some distance metrics over time. In addition, we integrate a Markov random field (MRF) motion model [33, 34] to model the interactions between the targets. The inclusion of the MRF in [33, 34] was originally used for modelling interactions between insects, where it was assumed that no two individuals were likely to occupy the same location at any given time. As the MRF has widely been used in image processing, we apply this concept in [35] to model target interactions where an MRF is constructed on the fly that penalizes those targets that are closely spaced with each other to avoid any track coalescence. Finally, to deal with the data association between the active targets and the available observations from the observers, we adopt the M-best 2-D assignment algorithm [6, 29– 32], where the top M-best assignments are used to approximate the marginal likelihood required in each particle weight calculation.


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The generalized state-space system for target tracking is given in this section. Denoting the combined state for Kt unknown and time-varying targets by xt ¼ fx1,t, . . ., xk,t, . . ., xKt, tg, we have the state evolution equation for xk,t [ Rnx, where k [f1,. . .,Ktg and nx is the number of elements in xk,t, as follows

but when Kt21 ¼ 0, we set hd ¼ 0 and hb ¼ h. Note the term hd/Kt21 when e Kt ¼ 21. This is present owing to the fact that the deletion of a target implicitly requires a random choice of which target from the current set of Kt21 should be terminated. Note also that this is a fairly crude model of target birth/death that does not include lifetime probabilities for individual targets. A more sophisticated model involving birth/death of individuals and explicit duration modelling of targets can be incorporated within our framework using fairly routine modifications to the dynamic prior (3). Tracking a variable number of targets requires either a variable dimensional state-space as in [16 – 19, 36], or a fixed dimensional state-space, say Kmax, in conjunction with a set of existence variables indicating which components of the state-space correspond to active targets [20]. While the second approach is easier to conceptualize as the dimension of the space is always fixed, in this paper we opt for the first approach of having a variable dimensional state-space which is more flexible in real-life applications. We do not impose a limit on the number of targets the proposed method can handle. Instead the proposed method allows the dimension of the state vector to adaptively change as the detected number of targets changes. In order to identify which components in xt belong to which tracks, a unique track label will be assigned to each track and the state vector, and will be discarded if the track to which it is assigned vanishes. Here, we define our parameters of interest as ut ¼ fxt, Ktg. Assuming that all targets evolve independently and according to Markovian dynamics [1, 37], we may express the prior distribution of ut as follows

xk;t ¼ f ðxk;t1 Þ þ vk;t ;

pðut jut1 Þ ¼ pðxt jxt1 ; Kt ; Kt1 ÞpðKt jKt1 Þ;

The rest of the paper is organized as follows. Section 2 presents the state-space model and the derivation of the required distributions, followed by a description of the MRF motion model for target interactions in Section 3. Section 4 presents the formulation of the sequential update for the target posterior distribution and the birth, death and update moves, respectively. Section 5 presents simulation results and performance evaluation, followed by the conclusions in Section 6. Finally, Appendix 3 introduces the necessary measures of performance (MoPs) for algorithm evaluation. Notations: Bold upper case symbols denote matrices, and bold lower case symbols denote vectors. The superscript T denotes the transpose operation, and the symbol ‘ ’ means ‘distributed as’. The quantity p(.j.) denotes a posterior distribution, whereas qa(.j.) denotes a proposal distribution function of parameter a. The notation (.)1:t indicates all the elements from time 1 to t. The quantity N(m, S) indicates a real normal distribution with mean m and covariance matrix S. The quantity U(a, b) indicates a uniform distribution over the interval [a, b], and UV indicates a uniform distribution within the volume V.

2. DATA MODEL 2.1.

Formulation of the problem and the data model

ð1Þ

where f(.), a linear or nonlinear state transition function, models the manoeuvring of the target. The quantities vk,t [ Rnx are assumed to be zero-mean Gaussian random variables with fixed and known covariance matrix Sv [ Rnxnx. For simplicity, we assume that only one target can appear and disappear at a given time and that Kt can be modelled by the following stochastic relationship Kt ¼ Kt1 þ [Kt ;

ð2Þ

where [ Kt is a discrete iid random variable such that the prior distribution function of Kt is 8 hd > < Prð[kt ¼ 1Þ ¼ Kt1 ; pðKt jKt1 Þ ¼ Prð[K ¼ 0Þ ¼ 1 hb hd ; t Kt1 > : Prð[kt ¼ 1Þ ¼ hb ;

where the first term in the above equation (4) is the dynamic prior function of xt, and the other term is the prior function of Kt. We can further expand p(xtjxt21, Kt, Kt21) YKt pðxt jxt1 ; Kt ; Kt1 Þ ¼ pðxk;t jxk;t1 Þ K¼1 8 QKt1 p0 ðxKt ;t Þ k¼1 pðxk;t jxk;t1 Þ; if Kt ¼ Kt1 þ 1; > : QKt1 if Kt ¼ Kt1 1; k¼1;k=k pðxk;t jxk;t1 Þ; where k* [ f1,. . ., Kt21g is the target that has vanished at time t and p0(xk,t) denotes the density of an initiated target which is often set to be uniform over the state-space, i.e.

ð3Þ

where hb, hd [ (0, 1), hb þ hd 1 are, respectively, the probabilities of incrementing and decrementing the number of targets. In general, we set hb ¼ hd ¼ h/2, where h [ (0, 1),

ð4Þ

p0 ðxÞ ¼ U X ðxÞ

ð6Þ

with X being the volume of the state-space where a new target may be initialized. Other forms may be used, if it is known, e.g. that targets are more likely to appear in certain regions of the state-space rather than others. In Section 4.3, we


MULTITARGET INITIATION, TRACKING present an effective approach for target state initialization that uses information from the observations to form regions of interest (ROIs) where targets may be proposed. Note that this independent form of prior dynamical model is modified to include dependencies between targets in Section 3. 2.2.

Observation model and likelihood

We consider No sensors independently operating and scanning with maximum range r nmax for n ¼ f1,. . .,Nog within a surveillance region Ry in the observation space. It is assumed that each observer’s motion is known1 at each time instant. We denote the position of the nth sensor at a given time by w nt [ Rnx. At any particular time, we receive a set of observations from No sensors, which are denoted by yt Wfy1t , . . ., y nt , . . ., yNt og, where y nt ¼ fy n1,t, . . ., y nM nt,tg denotes a set of Mnt measurements collected from the nth sensor. All observations in yt are the output of some sensor detection processes and assumed to be conditionally independent given the target state. Each measurement of ynt may originate from a true target or clutter. As in [37], if the mth measurement of ynt , i.e. ynm,t, originates from the kth target, it follows the model n n ¼ gðxk;t ; wtn Þ þ wm;t ; ym;t

n ym;t [ Rny ;

n jxk;t Þ ¼ N ð gðxk;t ; wnt Þ; Snw Þ; pðym;t

m [ f1; . . .; Mt g:

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a discussion of cases where a target may generate more than one measurement at a time, see [39]. Let A t be the space occupied by all feasible measurement-to-target assignments, and let lt ¼ fl1t , . . ., lNt og [ At be a set of measurement-to-target association stochastic hypotheses for No sensors, where the hypothesis with respect to the nth sensor is given by lnt ¼ fant , NCnt , NDnt g [40, 41]. It is further assumed that all hypotheses are mutually independent of one other. Given knowledge of Kt, the association n vector ant [ IMt is defined as

ant ¼ fan1;t ; . . .; anMtn ;t g;

am;t ¼

0; k;

if ynm;t originates from clutter; if ynm;t originates from the kth target

knk;t ¼

1; 0;

tract k is associated with a measurement; ð12Þ otherwise

with k¯k,t being the average value taken over all sensors, given by

ð8Þ

On the other hand, if the measurement is due to clutter, we will assume it to be uniformly distributed over Ry, i.e. m [ f1; . . . ; Mtn g;

ð9Þ

where V is the volume of Ry. However, given a set of measurements, it is not known which measurements are associated with which of the active targets and which are associated with clutter. Thus we need to properly deal with this data association uncertainty problem, i.e. measurement-to-target assignment, prior to target tracking. Two assumptions are made here for solving the data association problem. One is that each measurement can originate from either a true target or from clutter, and the other is that a target can produce at most one measurement at a given time. The first assumption means that the association is exclusive and exhaustive, whereas the second assumption implies that true targets may go undetected. For 1 In practice, the sensor state vector is usually provided by an on-board inertial navigation system aided by a global positioning system [38].

ð11Þ

and NCnt and NDnt are the number of clutter measurements and detected targets, respectively. We further denote an indicator of data association between track k and a measurement from sensor n at time t by knk,t, given as follows

k k;t ¼

1 n n pðym;t Þ ¼ U V ðym;t Þ¼ ; V

ð10Þ

where its mth element, for k [ f1, . . ., Ktg, is given by

ð7Þ

where ny is the number of elements in y nm,t, g(.) [ Rny may be a nonlinear function, and wnm,t [ Rny is a zero-mean Gaussian random variable with covariance Snw [ Rnyny. We can thus write the likelihood for a measurement due to a target as

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No 1 X kn : No n¼1 k;t

ð13Þ

It will be seen later that this indicator is useful in proposing track removal. Finally, we define a set of observation indices corresponding to the NDnt detected targets by Ia ¼ fm:anm,t ¼ k, k ¼ 1, . . ., Ktg. Given the association hypotheses lt, the total likelihood function for the observation set yt can be expressed as [40, 42] n

pðyt jxt ; Kt ; lt Þ ¼

Mt No Y Y

pðynm;t jxt ; Kt ; lnt Þ;

n¼1 m¼1

¼

8 No < Y n¼1

:

V

NCn t

Y l[I na

pðynl;t jxanl;t ;t Þ

9 = ;

ð14Þ :

Note that in practice, the number of feasible hypotheses in A t increases exponentially with an increase in the number of measurements and targets. As a result, the computational complexity required to deal with the data association uncertainty problem using classical methods, i.e. hypothesis


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enumeration and target-to-measurement assignment, becomes formidable with many targets and high-clutter rates.

2.3.

communication link to the central unit, the system is simple as there is only a single set of tracks to be maintained without any interaction among the sensors, and no risk of issues relating to data-reuse.

Association prior

We now present a prior function of lt ¼ (at, NCt, NDt), given the knowledge of Kt, which follows that described in [40, 43]. It is assumed that this prior function has the following hierarchical structure pðlt jKt Þ ¼ pðat jKt ; NCt ; NDt ÞpðNCt ÞpðNDt jKt Þ;

ð15Þ

where pðat jKt ; NCt ; NDt Þ ¼ Nl1 ðKt ; NCt ; NDt Þ; t

ð16Þ

pðNCt Þ ¼ ðLC ÞNCt expðLC Þ=NCt !;

ð17Þ

pðNDt jKt Þ ¼

Kt N PDDt ð1 PD ÞKt NDt : N Dt

ð18Þ

The prior of the association vector, conditional on the number of targets Kt and the number of clutter NCt and declared target measurements NDt, is assumed to be uniform over all the valid hypotheses as in (16), where the number of valid hypotheses is given by Nlt ðKt ; NCt ; NDt Þ ¼

NCt þ NDt N Dt

Kt ! : ðKt NDt Þ!

ð19Þ

3.

MRF MOTION MODEL

Here, we present a brief description of the MRF models employed, see [45] for further details. The idea is quite general and may be used to model many types of interaction betweeen targets, including both attraction forces for coordinated groups of targets and repulsive forces to prevent or discourage target collision. Here we adopt the latter mode, using the MRF in a simple repulsive mode, as in [33, 34], although current work is exploring much more sophisticated structures that encapsulate behavioural properties of group objects. An MRF is a graph with undirected edges E between nodes V, where the edges E specify a neighbourhood system and the nodes V represent random variables. The joint probability over the random variables is then factored as a product of local potential functions f at each node, and interaction potentials c defined on neighbourhood cliques. A commonly used form is a pairwise MRF, where the cliques are restricted to the pairs of nodes that are directly connected in the graph. Denoting the pairwise interaction potentials for targets i and j with states xi,t and xj,t by c(xi,t, xj,t), we modify the dynamic prior model as follows pMRF ðxt jxt1 ; Kt ; Kt1 Þ/ Kt Y

That is, all association vectors at are equally likely. It is assumed that the number of clutter measurements NCt follows a Poisson distribution (17) with expected value LC, which is assumed fixed and known. Finally, the binomial prior in (18) represents the number of target-originating measurements out of the total number of targets Kt, with a fixed and known PD, the target detection probability, shared by all targets.

2.4.

Data fusion for multisensor tracking

In this paper, we adopt a centralized tracking strategy [5, 44], in which all active No sensors in the network operate independently and send all information to a CPU to perform all necessary processing—global target initiation/termination, tracking and data association. In this case, all sensors essentially play the role of data collectors and do not perform local processing of any kind, other than any thresholding required to obtain the target detections at each time. It is the central node’s responsibility to initiate, maintain and terminate all tracks. While wide communication bandwidth is required as the observations from all sensors must be passed via the

k¼1

¼ c~ t

Y

pðxk;t jxk;t1 Þ

cðxi;t ; x j;t Þ;

i;j[f1;...;Kt g

Kt Y

pðxk;t jxk;t1 Þ;

k¼1

¼ c~ t pðxt jxt1 ; Kt ; Kt1 Þ;

ð20Þ

where c˜t is the overall interaction potential for all targets, given by

c~ t ¼

Y

cðxi;t ; x j;t Þ:

ð21Þ

i;j[f1;...;Kt g

The resulting dynamic prior function in (20) is essentially composed of two parts: the product of the predictive motion models in (5) and the MRF interaction potentials c(xi,t, xj,t) for all targets. It is the addition of the MRF interaction potentials that allows one to specify domain knowledge governing the joint behaviour of interacting targets. While other more complex formulations are possible to define the interaction potentials, pairwise potentials are here preferred because they are simple to specify and implement.


MULTITARGET INITIATION, TRACKING We express the pairwise potentials c(xi,t, xj,t) by means of negative exponential functions (unnormalized Gibbs distributions [46])

cðxi;t ; x j;t Þ ¼ expðfðxi;t ; x j;t ÞÞ;

ð22Þ

where f(xi,t, xj,t) is a penalty function that here penalizes those targets that are ‘close’ to each other. In other words, f(xi,t, xj,t) is maximal when two targets coincide and rapidly falls to zero as targets move apart. As a result not only can the target interactions be modelled, but also track coalescence problems can significantly be reduced. In short, to integrate the MRF dynamic prior with the framework, one simply needs to multiply the overall interaction potentials in (21) by the independent dynamic prior of (5). One important point is that now the dynamic prior is known only up to a constant of proportionality, see (20). This impinges on its use in SMC frameworks, where the dynamic prior may need to be evaluated pointwise in order to calculate weights. Here, however, we side-step this problem, making the assumption that the overall potential function is close to unity for very nearly all of the probability mass in the original dynamic prior p(xtjxt21, Kt, Kt21). This has been found acceptable in practice. Other schemes however could attempt to approximate the normalizing constants, or alternatively avoid the computation of the dynamic prior by drawing samples directly from the combined dynamical model itself (20); this could be achieved, e.g. using rejection sampling, with the unmodified dynamical model p(xtjxt21, Kt, Kt21) as envelope function, or by Markov chain Monte Carlo (MCMC) in more challenging cases where rejection sampling has low-acceptance rate.

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where all feasible association hypotheses are considered. The recursion is initialized with some distribution, say p(u0). It is well known that the update expression in (24) is analytically intractable for many models of practical interest. We therefore turn to SMC methods [12– 15, 47, 48], also known as particle filters, to provide an efficient numerical approximation strategy for recursive estimation of complex models. These methods have gained popularity in recent years, owing to their simplicity, flexibility, ease of implementation and modelling success over a wide range of challenging applications. In this section, we briefly describe the particle filter tracking framework for our state-space model.

4.1.

Sequential importance sampling

The basic idea behind particle filters is very simple: the target distribution is represented by a weighted set of Monte Carlo samples. These samples are propagated and updated using a sequential version of importance sampling as new measurements become available. Hence statistical inferences, such as expectation, maximum a posteriori estimates and minimum mean square error, etc., can be computed from these samples. N With a set of large N particles fu (i) t21gi¼1 with their associ(i) N ated importance weights fwt21gi¼1, we approximate the posterior distribution function p (ut21j y1:t21) at time t 2 1 as follows

p ðut1 jy1:t1 Þ

N X

ðiÞ wðiÞ t1 dðut1 ut1 Þ;

ð26Þ

i¼1

where d(.) is the Dirac delta function. We would like to genN erate a set of new particles fu (i) t gi¼1 from an appropriately selected proposal function, i.e. 4.

SMC METHODS

ðiÞ uðiÞ t qu ðut jut1 ; y1:t Þ

In the context of tracking, our goal is to estimate the posterior distribution of the parameters p(utjy1:t), which can recursively be obtained from two steps—prediction and update—according to the Bayesian Sequential Estimation framework in (23) and (24) as follows ð

p ðut jy1:t1 Þ / pðut jut1 Þp ðut1 jy1:t1 Þdut1 ; p ðut jy1:t Þ / pu ð yt jut Þp ðut jy1:t1 Þ;

X lt [At

pð yt jut ; lt Þpðlt jut Þ;

ð27Þ

(i) , . . ., k¯K(i)t21, t2 1g, and qx(.) and qk(.) are where k¯(i) ¯ 1,t21 t21 ¼ fk (i) the importance sampling functions of x(i) t and Kt , respectively. On the basis of the Markovian and independence assumptions, we may further expand qx(.), as given in (28),

ð23Þ ð24Þ

where the term p(ut21jy1:t21) in (23) is the posterior distribution function at t 2 1, and the term pu (ytjut) in (24) is referred to as the marginalized likelihood function, given by pu ð yt jut Þ ¼

ðiÞ ðiÞ ðiÞ ~ ðiÞ ¼ qxðxt jxðiÞ t1 ; Kt ; Kt1 ; y1:t ÞqkðKt jKt1 ; k t1 ; y1:t Þ;

ðiÞ ðiÞ qx ðxt jxðiÞ t1 ; Kt ; Kt1 ; y1:t Þ 8 q0 ðxK ðiÞ ;t jy1:t Þ > > t > > ðiÞ > QKt1 ðiÞ ðiÞ > < if KtðiÞ ¼ Kt1 þ 1; k¼1 qx ðxk;t jxk;t1 ; y1:t Þ; ð28Þ ¼ QK ðiÞ ðiÞ t > qx ðxk;t jxðiÞ ; y1:t Þ; if KtðiÞ ¼ Kt¼1 ; > k¼1 k;t1 > > > ðiÞ > : QKt1 ðiÞ q ðx jxðiÞ ; y Þ; if KtðiÞ ¼ Kt1 1; k¼1;k=kdðiÞ x k;t k;t1 1:t

ð25Þ (i)

where k d [ f1, . . ., K(i) t21g is the target that has vanished at


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time t and q0(xk,tjy1:t) the proposal function for a newly inititated target and qx(.j.) the proposal function for a continuing active target with state xk,t. When sampling K(i) t from qk(KtjK(i) ¯ (i) t21, k t21, y1:t), we may propose a death move based on k¯(i) t21 for track removal, a birth move based on detected activities using y1:t for track initiation, or an update move for track maintenance. We will present the detailed formulation of these proposal functions in subsequent sections. With the set of particles fu(i) t g obtained from (27), we may then update the associated importance weights w(i) t as follows ðiÞ wðiÞ t / wt1

ðiÞ ðiÞ pu ðyt juðiÞ t Þpðut jut1 Þ ðiÞ qu ðuðiÞ t jut1 ; y1:t Þ

ð29Þ

PN (i) with i¼1 wt ¼ 1, It follows that the new set of particles N N fu(i) g with the associated importance weights fw(i) t i¼1 t gi¼1 is then approximately distributed according to p(utjy1:t). We note that this form of particle filter explicitly propagates a set of particles having variable dimensionality (number of targets). Other work where variable dimension particle filtering occurs is found in [22, 36]. As the particle filter operates through time, only a few particles contribute significant importance weights in (29), leading to the well-known problem of degeneracy [14, 47]. To avoid this, one needs to resample the particles according to their importance weights. That is, those particles with more significant weights will be selected more frequently than those with less significant weights. More detailed discussion of degeneracy and resampling may be found in [14].

4.2.

Proposal mechanism for particle filter

In any particle filtering setup, and in particular for this variable dimension setting, the proposal function qu(utju(i) t21, y1:t) in (27) forms a crucial part of the design. In particular, we here have to design proposals for initiation of new targets (when Kt ¼ Kt21 þ 1) and maintenance of existing target tracks. The proposal function will need to be data dependent if we are to avoid the need for huge numbers of wasted Monte Carlo samples, generated in unlikely parts of the state-space. There are many ways to construct such data-dependent proposals. For example, see [36] for data-dependent proposals in a track-before-detect setting. Here, within a detection-based setting, we perform clustering of observations over time (‘intra-sensor ROIs’) and over space (‘inter-sensor ROIs’) in order to locate likely regions for target motion and initiation. Conversely, termination moves are proposed for targets that are not associated with observation clusters. The motivation here is that observations will tend to cluster around places where targets exist, either spatially or temporally, while clutter is on average less likely to form such clusters. Such a procedure is of course not 100% reliable, and we will resort to the particle weights accumulated over time to resolve

which clusters are indeed target-originating, and which are not. Appendix 1 gives details of how ROIs are formed from the multiple sensor data; we note that for the single sensor case, the temporal clustering methods used for intra-sensor ROIs have already been described in [19, 29, 31], whereas the inter-sensor ROI method is similar in concept to that in [49, 50]. 4.3.

Target initiation, termination and track maintenance

Here we extend the target initiation and deletion method proposed in [19, 29, 31], in which the number of targets are deterministically computed based on ROIs, to an approach where for each particle we decide randomly whether to initiate a new track, terminate an existing track or keep Kt unchanged from the previous time. For each particle, these three types of move are selected with probabilities hb, hd and 1 2 hb 2 hd, respectively, in accordance with the dynamical model for Kt, see (3). As in [19, 29, 31], measures of persistence in ROIs are used as a means to decide where initiation of a target should occur. 4.3.1. Initiation move Once a birth move has been decided upon for a particular particle, all available ROIs computed at that time are examined and a persistence measure is computed for each, based upon the number of observations that have been placed in that ROI over several time frames (intra-frame) and over all sensors at the current time (inter-frame). The inclusion of a term involving inter-sensor clusters helps to guard against false alarms in detection, but can cause a delay in the detection of a true track. In the results section, we compare the use of intra-frame clusters alone to determine persistence (‘Condition 2’) against the combination of inter- and intra-frame clusters (‘Condition 1’). ROIs that have already been associated with a target in that particular particle are not considered to be available for initiation purposes. One particular ROI is now selected randomly from the available set of ROIs for initiation of the new target, with probability proportional to its persistence measure. 4.3.2. Termination move In a similar fashion, when a termination move has been selected the past history for each existing target in a particular particle is examined and a persistence metric is calculated which measures how infrequently each target has been associated with a ROI over recent time slots. The target for termination is then selected randomly with probability proportional to this persistence metric. 4.3.3. Update move for surviving targets Once all births and deaths have been decided upon for all particles as described above, the particles fK(i) t g are updated appropriately to have the correct dimensionality, by deleting


MULTITARGET INITIATION, TRACKING terminated paths and initiating new targets where a birth move has occurred. It then remains to sample state values for new targets and to update the states for targets that continue from the previous time step. A key element in generating a set of weighted particles to suitably approximate the posterior distribution function in (24) is the selection of the proposal importance sampling func(i) (i) (i) tion to generate samples, i.e. x(i) t qx(xtjxt21, Kt , Kt21, y1:t) in (28). Here, we propose a data-dependent importance sampling strategy for individual targets k ¼ f1, . . ., K(i) t g that involves the recent observation information contained in ROIs St [31, 51]. Many choices would be workable within our framework. Here, we adopt a simple mixture of the prior dynamics and a data-dependent proposal based on ROI information: ðiÞ ðiÞ xðiÞ k;t qx ðxk;t jxk;t1 ; y1:t Þ ¼ ð1 mÞpðxk;t jxk;t1 Þ ðiÞ þ mqx ðxk;t jxðiÞ k;t1 ; X k;t1 ; S t Þ; 0 m 1;

ð30Þ

(i) t21 where X (i) k,t21 ¼ fxk, t0 gt0 ¼t2 tx collects a buffer of tx past states of track k up to time t 2 1. Details of construction of the selected proposal function can be found in Appendix 2.

4.4.

Target state estimation

(i) We now have N particles fu (i) t g with associated weights fwt g, representing the variable dimension target posterior distribution. In principle, we can now estimate the target state xˆt for all active targets. In practice, this is not so straightforward, since there could be labelling ambiguities and track-swapping issues between different particles in the representation. In PHD filtering approaches, it is suggested [16–18] that spatial clustering methods can be used to group closely spaced state samples to compute the target state estimates. See [52] for further duscussion concerning the labelling ambiguity problem. Here, we adopt a simple and workable sub-optimal scheme that associates a unique label with each track in each particle, obtained from the identity of the ROI which generated that track. For all particles, we extract all track components which have a given track label, and then compute a weighted sum of this group of particles for the track. For example, consider an active track with label tra. Denote by Itra a set of particle indices that contain the state components of track tra. Then we have,

AND

b xtra ;t

X i[I tra

ðiÞ w~ ðiÞ t T½xt ; tra ;

w~ ðiÞ t ¼ P

wðiÞ t

;

ð31Þ

681

Approximate likelihood calculations via M-best data association

4.5.

The marginalized likelihood of (25) is required in weight computation for the particle filter [see (29)]. This calculation requires a full enumeration over all possible hypotheses, which becomes infeasible for large numbers of targets and clutter points. One attractive option is a soft-gating approach, in which the particle state vector is augmented to include a randomly sampled association vector in addition to target state variables [53, 54]. An alternative is to approximate the summation by computing only the most probable components, an approach we adopt here, based on auction algorithm solutions to the 2-D assignment problem. The 2-D assignment algorithm [5, 6] is an intuitive method for solving classical assignment problems, which includes the data association problem for MTT applications, given that the assignment is always on a one-to-one basis. In every iteration, this approach globally searches for the best feasible solution that minimizes a 2-D cost function, subject to a set of constraints. Methods such as the auction algorithm [55 – 58], Munkres algorithm [59], Jonker, Volgenant and Castanon algorithm [60] and signature methods [61] may be used to compute the best feasible assignment. In this paper, we choose to use the auction algorithm because of its favourable computational requirements and optimality of solutions [6]. To mitigate the drawback inherited from the hard irrevocable decisions provided by the best-solution approaches, it is suggested [5, 6] that the M-best assignment algorithm be used that provides a set of feasible solutions with their own probabilities. These feasible solutions will subsequently be used to approximate the full marginalized likelihood function in (25) [29, 30, 62] in which these assignments are taken into account to appropriately weight the posterior distribution function. N Given a set of particles fu(i) t gi¼1 and the current observations yt, we need to compute a set of assignments for every observation vector ynt for n [ f1, . . ., Nog. These assignments and the associated probabilities for observation ynt are denoted PMby n (i) (ij) (ij) (i) t and U(ant ) for j ¼ f1, . . ., Mnt g, where ant j¼1 n (ij) n (ip) n (iq) U(at ) ¼ 1 and U(at ) U(at ) for p , q and p,q ¼ f1, (i) . . ., Mnt g. Accordingly, we may obtain a set of data associ(i) (ij) ation hypotheses lnt ¼ flnt g for observer n, where the jth hypothesis particle is given as follows

lnt wðiÞ t

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ðij Þ

ðij Þ

ði Þ

ði Þ

¼ fant ; NCnj ; NDjnt g; t

ðiÞ

j ¼ f1; . . . ; Mnt g;

ð32Þ

where

i[I tra

nx1 where T[x(i) extracts the state component of track tra t , tra] [ R in particle i.

ði Þ NCnj t

Mtn X ði Þ ði Þ ði Þ ¼ d am;tj and NDjnt ¼ Mtn NCnj : m¼1


t

ð33Þ

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W. NG et al. (i)

The number of hypotheses Mnt to be retained is different for every particle i and sensor n and is determined as follows ði

Þ

Uðant M Þ 0:01; PM nði j Þ j¼1 Uðat Þ

ð34Þ

where the Mth hypothesis is retained if and only if it is significant (say 1%) when compared with the culmulative probability. After all M hypotheses are obtained, their weights will be normalized to one. Once we have obtained all No sets of data association n (i) hypotheses l(i) t ¼ flt g for n ¼ f1, . . ., Nog, we can then substitute them to approximate the marginalized likelihood function in (25), i.e. nðiÞ

pu ðyt juðiÞ t Þ

No M t Y X

ðij Þ

ðij Þ

ðiÞ n n pðyt juðiÞ t ; lt Þpðlt jut Þ

ð35Þ

n¼1 j¼1

and then update the importance weight according to (29).

5. COMPUTER SIMULATIONS In this section, we examine the performance of the proposed algorithm in initiation and tracking three targets appearing and disappearing at different times during the experiment. In particular, the trajectory of each target is synthesized by the following model [63] xk;t ¼ f ðxk;t1 ; vk;t Þ þ nk;t ; 2 sin vk;t DT 1 cos vk;t DT 3 1 0 6 7 vk;t vk;t 6 7 60 7 cos v DT 0 sin v DT k;t k;t 6 7xk;t1 þ nk;t ; ¼6 7 1 cos v DT sin v DT k;t k;t 60 7 1 4 5 vk;t vk;t 0 sin vk;t DT 0 cos vk;t DT ð36Þ where DT is the sampling instant and vk,t (2 p, p] is the turnrate parameter. In the experiments, the sensors are assumed to be stationary with known positions. Moreover, the observation model for mth measurement of ynt from the nth observer is a nonlinear model with bearing and range measurements, given as follows " ynm;t ¼ " ¼

unk;t n rk;t

# þ wnm;t ;

tan1 ðxk;t xnt =ykt ynt Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxk;t xnt Þ2 þ ðykt ynt Þ2

ð37Þ

# þ

wnm;t

with unk,t and rnk,t being the true bearing,2 and (xk,t, yk,t) and (xnt , ynt ) are the xy coordinates of target k and observer n, respectively. The state and observation noises nk,t and wnm,t are white Gaussian with zero mean and covariances Sn and Snw, respectively, defined as "

# " 2 s2x Sx 022 snu n ; S ¼ Sn ¼ w 022 s2y Sy 0 3 DT =3 DT 2 =2 Sx ¼ Sy ¼ ; DT 2 =2 DT

#

0 2

snr

; ð38Þ

where 022 is a matrix of zeros, and sx ¼ 0.1 and sy ¼ 0.1 are the state noise standard deviations. For simplicity, it is further assumed that all sensors share identical maximum range rmax ¼ 2000 and number of expected clutter, i.e. Ln ¼ LC8n, and that all measurements collected at different sensors share identical noise. To evaluate the performance of the proposed algorithm, we introduce MoPs (see Acknowledgements section) suitable for a data fusion environment. These MoPs are completeness (MoP1), continuity (MoP2) and kinematic accuracy(MoP3), which are defined in Appendix 3. In addition to these MoPs, we also compute Pdet and e K, which are the probability of true targets that are correctly detected and the average target detection error out of L independent trials in a variety of environments, respectively. To obtain Pdet, we essentially count the number of times every true target can be estimated and maintained by a track with its completeness larger than 80% (see the definition of MoP1) over L runs. Therefore, if five out of six true targets in a scenario satisfied the above conditions, then Pdet would be 83%. To obtain e K, which is essentially the Root Mean Square Error (RMSE) of estimating Kt, we use vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u L X T u1 X eKt ¼ t jK 0 K^ tl j2 ; LT l¼1 t¼1 t

ð39Þ

where K0t is the number of true targets and Kˆlt is the number of established tracks in the lth trial, respectively.

5.1.

Experiment 1: Joint target initiation/deletion and tracking manoeuvrable targets

In this experiment, we demonstrate the performance of the proposed method for joint target initiation and tracking for three synthesized targets. These targets appear and disappear at different times. Table 1 gives a description of these synthesized targets whose trajectories are synthesized according 2 The true bearing angle is measured from the north (N) or south (S) ends of the reference meridian [38].


MULTITARGET INITIATION, TRACKING

AND

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TABLE 1. Descriptions of the targets in Experiment 1. Descriptions

Target 1

Target 2

Target 3

Time duration Initial position Initial velocity

[61, 261] [260.7, 361.5] [25, 25]

[24, 250] [320.0, 239.4] [25, 5]

[34, 253] [212.5, 508.7] [5, 25]

to the dynamical model in (36) with DT ¼ 1 s. We randomly generate the turnrates vk,t governing the manoeuvres of each target as piecewise constant whose durations and switching times are also randomly synthesized. Figures 1 and 2 depict the synthesized turnrates and trajectories of these targets, respectively. Not only are these targets highly manoeuvrable, but they also make crossings with each other at various times. Three stationary sensors are deployed at w 1 ¼ (400, 400), w 2 ¼ (0, 400) and w 3 ¼ (0, 2400). In this experiment, we compare the performance of the proposed method under different scenarios summarized in Table 2. In these scenarios, the number of expected clutter points per sensor, LC, varies, while maintaining fixed standard deviations for the observation noise (su ¼ 1.58, sr ¼ 15) and target detection probability PD ¼ 0.5. A total of T ¼ 300 observations is generated according to (37). An example of the synthesized observations for Scenario 2 is shown in Fig. 3. In addition, in every scenario, we also evaluate the performance of target initiation when using different versions of the birth persistence measure: Condition 1 uses a persistence measure based solely upon intra-frame clustering, whereas Condition 2 incorporates also a term involving inter-frame clusters (see discussion in Section 4.3.1.). For each scenario, L ¼ 30 independent trials are conducted in which in every trial, the observations are re-generated while the target

FIGURE 2. The synthesized target trajectories in Experiment 1 with the times at which they appear and disappear.

trajectories remain intact. Figures 4 – 6 and Table 3 exhibit the results of this experiment. The number of particles used in every trial is N ¼ 500. Figure 4 shows the results of target initiation and deletion for different scenarios and conditions. It can be seen that when LC is small the results from both conditions are comparable as the likelihood of initiating spurious tracks is small, but the delays taken for track initiation under Condition 2 are generally shorter than those under Condition 1 (see Figs. 4a and b and Table 3). Nevertheless, it is not difficult to see that the proposed method under Condition 2 is more susceptible to false track initiation (see those times at which no true targets exist). On the other hand when LC increases so do the number of false tracks from the proposed method and the estimation error under both conditions. It is clear that the proposed method under Condition 1 outperforms Condition 2 in reducing the initiation of false tracks as evident in Fig. 4c and the RMSE in Table 3. In summary, the proposed method with the inclusion of inter-frame clusters in the birth persistence measure (Condition 1) performs more consistently and is less sensitive to the number of clutter measurements in target initiation for all scenarios (Fig. 5). The reduction of the number of spurious tracks is, however, achieved at the expense of relatively longer delays in track initiation.

TABLE 2. Parameters for different scenarios in Experiment 1.

FIGURE 1. The synthesized turn rate trajectories of the targets in Experiment 1.

Parameters

Scenario 1

Scenario 2

Scenario 3

su, sr No LC

1.58, 15 2 (w 1, w 2) 5

1.58, 15 3 (w 1, w 2, w 3) 10

1.58, 15 3 (w 1, w 2, w 3) 15


684

W. NG et al. the target has made the turn. Table 3 gives a quantitative evaluation on tracking performance MoP3 of the proposed method. Under the same scenarios, we also compare the proposed method with the Existence Joint Probabilistic Data Association Filter (EJPDAF) [20] in terms of joint target initiation/ deletion and estimation. EJPDAF, an extension of JPDAF [1, 64], offers a full joint treatment of the state and target existence variables using particle filters, enabling a practical strategy for target initiation, maintainence and termination. The comparison results are summarized in Table 4. According to Table 4, the performance between these two methods is comparable in Scenario 1, which is characterized with smaller levels of noise and clutter points. When these levels go up in Scenarios 2 and 3, the proposed method outperforms EJPDAF. Since the EJPDAF does not have an explicit procedure to handle the time-varying state transition model in (36) and uses a fixed NCV dynamical model throughout the target estimation, its performance may be degraded due to the mismatch between the NCV model and the one used in the simulation. As a result, the performance in target initiation/deletion and estimation is adversely affected. Although all true targets are detected, the number of false (spurious) tracks initiated is larger than that by the proposed method, probably due to the large number of clutter measurements per scan. This finding is consistent with that in [20]. In addition, as the target dynamical prior functions are chosen as the target state proposal functions, the samples so-generated are inefficient and have insignificant weights, further degrading the tracking performance.

5.2.

FIGURE 3. The synthesized observations (bearing and range) for all sensors in Experiment 1 in Scenario 3.

Figure 6 shows a comparison between the true and the mean estimated target trajectories over L trials in this experiment for the scenarios. It can be found that the proposed method performs well in tracking these manoeuvrable targets for all scenarios in the absence of multiple dynamical models. Furthermore, almost all the estimation error ellipses with 95% CI encapsulate the true target trajectories. In general, the tracking performance improves when more sensors are deployed. For example, in Fig. 6d – f, one can see that a faster response to the turn made by target 2 at about (2200, 0) is achieved when three sensors are deployed (Scenarios 2 and 3), whereas the tracker with two sensors in Fig. 6d needs to take a more time steps to adapt to the change after

Experiment 2: Evaluation of MRF in dealing with target interactions

In this experiment, we focus on the evaluation of the performance of MRF motion model in dealing with the interaction between targets, especially when they are closely spaced. Table 5 gives a description of these synthesized targets whose trajectories are once again synthesized according to (36) with randomly generated turn rates as in the last experiment (see Fig. 7). The target trajectories are designed such that these targets are widely separated inititally and approach each other until point A and then separate again. These targets will then get very close to each other at points B and C, and eventually separate. Table 6 lists the distances between these two targets at these points. Depending on the noise level, the points A, B and C can lead to severe ambiguities in target tracking such as track swapping and merging. Table 7 lists the parameters used in two scenarios, each of which is characterized with different noise levels with expected number of clutter LC ¼ 10 and target detection probability PD ¼ 0.5. A total of two stationary sensors are deployed at (400.0, +400.0) in this experiment.



AND

TERMINATION

685

FIGURE 4. A comparison between the true and the estimated number of targets in Experiment 1 for all scenarios and conditions (Table 2 and Conditions 1 and 2), averaged over 20 independent trials.

Figure 8a and b shows an example of the target originating measurements for each scenario in Table 7. It can be seen that when the noise level is low the ambiguites at A – C are low (Fig. 8a). On the other hand, when the noise level becomes higher the ambiguites become severe (Fig. 8b) and it appears that the targets had crossed at point A, leading to track swapping and possibly track coalescence in the remaining portion of tracking. Here, we examine the performance of the MRF model in track maintainence where targets interact with N ¼ 500 particles. A total of L ¼ 30 independent trials are conducted for each scenario. The figures in Fig. 9 exhibit the evaluation results using the proposed method with and without MRF. Figure 9a and b shows a comparison between the true

tracks and their mean estimates over L ¼ 30 trials for Scenario 1. While both methods are able to perform well without suffering from any track swap or coalescence problem, one can see that the tracks estimated without MRF appear to attract to each other at point A (Fig. 9b). When the observation noise becomes higher as in Scenario 2, the track swap and coalescence problems become more severe that degrade the tracking performance of the method without MRF as evident in Fig. 9d. On the other hand, the method with MRF demonstrates its robustness in handling close target interactions under the same noise conditions and clearly outperforms the other method as evident in Fig. 9c. Table 8 summarizes the quantitative evaluations in this experiment.


686

W. NG et al. evaluated the performance of the proposed method under various conditions in terms of appropriately chosen MoPs.

ACKNOWLEDGEMENTS The authors appreciate QinetiQ to provide valuable information of performance metrics for algorithm evaluation.

FUNDING The work of WN, JL and SG was funded by the UK Government’s Data and Information Fusion Defence Technology Centre (DIF-DTC) under project 10.2.

7.

FIGURE 5. A comparison of the performance of target initiation and termination for all scenarios under the conditions over L ¼ 30 trials.

6. CONCLUSIONS In this paper, we presented a simulation-based approach of online joint detection and tracking for multiple targets using multiple sensors. We also integrated an MRF-based motion model in the framework to model the close target interactions. A central-level data fusion strategy was used to collect all observations from the active sensors, followed by the maintenance of a set of global tracks using SMC methods. Computer simulations demonstrated that the proposed approach performed well in jointly detecting and tracking time-varying number of manoeuverable targets in a highly cluttered environment with low target detection rate. Moreover with the integration of MRF, the proposed method gained significant improvement in track maintenance when targets moved in close proximity to one another. We also

APPENDIX 1: DETAILS OF INTRA-SENSOR AND INTER-SENSORS CLUSTERS

Suppose an activity has appeared in the surveillance region, and the observations originating from this activity may be grouped over time by the deployed sensors. Two types of observation clusters can be formed: intra-sensor clusters and inter-sensor clusters. The former are constructed from nearby observations from a sensor over time, whereas the latter are from nearby observations from all deployed sensors at a given time. Denote a set of intra-sensor clusters by St for all available sensors, and denote the jth intra-sensor cluster by S nj #S t, containing observations collected from sensor n [ f1, . . ., Nog. On the other hand, denote a set of inter-sensor clusters at time t by y Ct , consisting of a subset of observations from all sensors that are closely spaced. In the following example, No ¼ 4 sensors are used. Figure A1 is an illustration of a set of inter-sensor clusters yCt . In this particular example for No ¼ 4 sensors, three clusters yCt l for l ¼ fA, B, Cg are located. Note that no two measurements from the same sensor will be grouped in an inter-sensor cluster. Other things being equal it can be said that cluster C is more likely to have originated from a true target than clusters A and B in terms of the number of sensors that are included in a cluster. Figure A2 is an illustration of the presence of both intersensor and intra-sensor cluters. An activity is captured by a set of Snj for n ¼ f1, 2, . . ., 4g from t 2 3 to t. These intrasensor clusters are represented by the lines in the figure, and each line connects different number of measurements within this short window of time. For example, the cluster from sensor 4 has only two points, whereas that from sensor 1 has four points. Accordingly, cluster 1 will have a higher chance to be selected for track initiation than the other clusters since it has more consecutive measurements. In general, if a given ROI, say Snj , has never been used for track initiation, or if it has not been associated with any active track particle



AND

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687

FIGURE 6. A comparison between the true and the estimated target trajectories in Experiment 1 for all scenarios. The solid lines represent the true target trajectories whereas the lines with dots represent the estimated trajectories with the 95% error ellipses.

i at time t, this ROI is a potential candidate to propose a new track. However, the activity represented by this ROI may originate from either a true target or a spurious object, and it may be premature to initiate a new track solely based on this ROI from a single sensor. Where No sensors are simultaneously operating, we may raise the confidence level when a track is initiated using a persistence measure as follows

where Ncm(Snj ) is the number of links connecting consecutive measurements included in Snj and NC(Snj > yCt ) is the number of measurements included in the inter-sensor cluster in yCt that overlaps with the ROI Snj . The quantity nb in (A.1) is a constant TABLE 3. Evaluation results for Experiment 1 with L ¼ 30 independent trials of the scenarios in Table 2. Scenario Pdet (%) e K

T Ncm ðS nj Þ NC ðS nj yCt Þ ;1 ; pnj ¼ min nb No

ðA:1Þ

1 2 3

96.0 95.0 90.0

MoP1 (%) MoP2 (%) MoP3 (1 2 s)

0.1372 94.1 0.0892 92.2 0.2084 87.3


7.9 8.4 9.5

23.1 (4.7) 18.73 (2.1) 20.2 (3.3)

688

W. NG et al.

TABLE 4. Evaluation results for Experiment 1 with L ¼ 30 independent trials of the scenarios in Table 2 for the EJPDAF. Scenario

Pdet (%)

eK

MoP1 (%)

MoP2 (%)

MoP3 (1 2 s)

1 2 3

90.0 93.0 86.0

0.12 0.24 0.33

91.6 88.3 85.0

8.0 14.4 18.6

13.4 (4.2) 11.7 (4.3) 18.2 (5.2)

related to the expected number of consecutive measurements required for establishment of a new target. In short, a new track is only initiated iff an activity has demonstrated persistence over a window of time within an individual sensor as well as among the majority of deployed sensors at a given time. A numerical example of how to obtain pnj is given in Appendix 1. In practice, we first compute the persistence measures for all unassociated ROIs fpnj g and then sample one most likely ROI for track initiation for every particle i 8i. According to the definition of pnj in (A.1), we need to obtain Ncm(Snj )/nb and NC(Snj > yCt ). In this example for each sensor n, we need to first count how many links connecting consecutive points up to time t in each line as shown in Fig. A2. For instance, sensor n ¼ 1 has four consecutive measurements connected, whereas sensor n ¼ 2 has only two from t 2 3 to t. We then count how many points in yCt overlapping with the available fSnj g. This involves answering two questions: (1) does an intra-sensor cluster overlap with an inter-sensor cluster by sharing a common measurement at the same time? and (2) if they do share a common measurement how many measurements are included in that inter-sensor cluster? For instance, the inter-sensor cluster at time t overlaps with the intra-sensor clusters from all sensors except n ¼ 3, and has three points or sensors included within. Accordingly, we have No birth probabilities of the available intra-sensor clusters as shown in Table 9. As a result in this example, intrasensor clusters from sensor n ¼ 1 and n ¼ 2 are more likely to be selected for target initiation at time t.

FIGURE 7. The synthesized target trajectories in Experiment 2.

information of the target, and we express x (i) ¼ fs (i), s˙(i)g, where s (i) and s˙(i) are the xy positions and the velocities of the initialized target. Assume that every measurement in Snj contains position information of an object, we propose to adopt a simple linear regression model to initialize the s (i) and s˙(i). The steps are described as follows. (i) Let ROI Snj be chosen for target initiation, (a) We first map the measurements fzl, tlg with their time points (tl . tlþ1) enclosed in Snj onto the corresponding xy position according to the inverse mapping i.e. fl ¼ g 21(zl, wnt ), where g(.) is defined in (7). (b) Assume that Lj recent observations in Snj are chosen for initialization, we can express the linear regression model using fflg and ftlg for l ¼ f1, . . .,Ljg as follows fl ¼ s s_ ðtLj tl Þ þ el ; 8l;

where the error e l is assumed to be a zero-mean, Gaussian random variable with covariance matrix Se l. (c) Using the standard least-squares estimation, we may then obtain x^ ¼ f^s; s^_g, as well as the covariance matrix of the estimation error Se l. (ii) Sample x (i) N(xjxˆ, Se l) for target state parameter initialization.

8. APPENDIX 2: DETAILED FORMULATION OF IMPORTANCE FUNCTION OF TARGET STATE 8.1.

ðA:2Þ

Target state initialization

After a new track is initiated with an ROI Snj , we may use the information in that ROI to initialize the new target state. Let fx (i)g [ RX be a target state particle of a new track. A typical target state vector contains position and velocity TABLE 5. Descriptions of the targets in Experiment 2. Descriptions

Target 1

Target 2

Time duration Initial position Initial velocity

[6, 241] [564.2, 21007.0] [25, 25]

[28, 249] [848.1, 2466.2] [25, 5]

TABLE 6. The distance between two targets at A, B and C in Experiment 2. A

B

C

50.0

25.0

60.0


MULTITARGET INITIATION, TRACKING TABLE 7. Parameters for different scenarios in Experiment 2. Noise parameters

Scenario 1

Scenario 2

su, sr

18, 10

38, 10

8.2.

We may find an association between target k and some available ROIs in St and then use the observation information in these associated ROIs to generate state particles. Since it is seldom that measurements contain velocity information of targets, we first propose to generate velocity samples of a target from its past positions. (i) (i) To generate samples x (i) k,t qx(xk,tjx k,t21, X k,t21, St) in (30), (i) t21 (i) where X k,t21 ¼ fx k,t0 gt0 ¼t2 tx is a buffer of tx past states of target k, we adopt a proposal procedure based upon approximate linearization and the Extended Kalman Filter [14, 47], where the point at which a first-order Taylor series expansion, say x(i) o , is approximated from the past state trajectories. Let Snj #St be one of the associated ROIs with target k. The steps to generate the particles using the information in X(i) k,t and S nj are given as follows. (i) For particle i, we first generate a sample of x(i) o from as follows. X (i) k,t21 (i) (a) Let s˜(i) t0 ¼ Px k,t0 with 1 P¼ 0 where

s˜(i) t0

0 0

0 0 ; 1 0

contains xy position of

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(b) Express the linear regression model as follows _ ðt0 t 1Þ þ et0 ; s~ ðiÞ t0 ¼ s s

8t0;

ðA:4Þ

where the error e t0 is assumed to be a zero-mean, Gaussian random variable with covariance matrix Se t . (c) Compute the least-squares estimate x^ ¼ f^s; s^_g, as well as Se t , and ultimately generate a sample ˆ , Se t . x(i) o N(xojx (ii) We then take the most recent measurement in Snj , say z, to generate x(i) k,t as follows. 0

Importance sampling function of x (i) k,t

AND

0

0

(a) Perform a first-order Taylor series expansion of the observation function g(.) in (7) around the particle (i) . f (i) o ¼ f(x o ) with the Jacobian function G of g( ) . evaluated at f (i) o (b) Draw samples according to a Gaussian approximation of the proposal of the form [14, 47] ðiÞ ðiÞ xðiÞ k;t N ðxk;t jmt ; St Þ;

ðA:5Þ

1 1 T 1 SðiÞ t ¼ ðSv þ G Sw GÞ ;

ðA:6Þ

ðiÞ ðiÞ T 1 S1 mðiÞ t ¼ St v f o þ G Sw Dz ;

ðA:7Þ

n ðiÞ Dz ¼ z gðf ðiÞ o ; wt Þ þ Gf o :

ðA:8Þ

where

ðA:3Þ

x(i) k,t0 .

FIGURE 8. An example of the target originating measurements for the scenarios in Experiment 2.


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FIGURE 9. A comparison between the true and estimated targets (averaged over L ¼ 30 independent trials) with and without MRF to model the target interactions in Experiment 2.

In the case when more than one ROI is associated with target k because of the deployment of multiple sensors, one may take an average of these individual samples.

TABLE 8. Evaluation results for Experiment 2 with L ¼ 30 independent trials the scenarios in Table 7. Methods

MoP3 (1 2 s)

Track swap (%)

With MRF Without MRF

25.1 (5.6) 58.1 (7.9)

5.0 30.0

9.

APPENDIX 3: DEFINITIONS OF MOPS

Here we give the definitions of four MoPs provided by the authors’ sponsors (see Acknowledgements section) that are used to provide a quantative evaluation on a given target tracking algorithm. These MoPs are completeness, continuity, kinematic accuracy and clarity, and their definitions are given as follows. (i) MoP1 completeness: The measure of the portion of true objects that are included in the estimated tracks. Let T ok ¼ ft0k , t1k g be the times at which true target k appears and disappears, and T ek ¼ ft˜0k , ˜t1k g be the



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TABLE 9. The birth probabilities of the ROIs in Fig. A2

FIGURE A1. A graphical illustration of inter-sensor clusters for No ¼ 4, where every number n represents a measurement received by sensor n at time t.

times at which the estimated track for target k is initiated and terminated. Completeness for target k is then Completenesslk ¼

Sensor

pnj

n¼1 n¼2 n¼3 n¼4

3/4 min(1, 3/4 min(1, 0/4 min(1, 3/4 min(1,

(ii) MoP2 continuity: The measure of how accurately the estimated tracks maintains track numbers over time. For example, the estimated track is continuous when the track number assigned to an object does not change. That is, when a true target is associated with more than one track, track ‘loss’ problem may have occured and almost all these associated tracks will have termination times smaller than the true time t1k . Therefore, to evaluate this metric, we may count the number of tracks with which a true object is associated, nlk 1, over all targets in a given scenario, as follows

~t1k ~t0k þ 1 j~t0k tk0 j j~t1k tk1 j tk1 tk0 þ 1

jnK Kmax j 100%; MoP2 ¼ 1 n K

100%: ðA:9Þ That is, the closer Completenesslk to 100%, the better is the performance of the tracker. For all tracks and runs, we define completeness as follows MoP1 ¼

Kt L X 1 X Completenesslk : LKt l¼1 k¼1

ðA:10Þ

3/nb) 1/nb) 2/nb) 0/nb)

ðA:11Þ

where Kmax is the maximum number of true tracks in a scenario and n¯K is n K ¼

Kmax L X 1 X nl : L Kmax l¼1 k¼1 k

ðA:12Þ

(iii) MoP3 kinematic accuracy: The measure of how accurately a system reports track position and velocity. The estimated track is kinematically accurate when the position and velocity of each assigned track agree with the position and velocity of the associated object. An example of this measure is the RMSE. To evaluate kinematic accuracy, we adopt the RMSE for L independent runs, defined by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u L u1 X RMSE2l MoP3 ¼ RMSE ¼ t L l¼1

ðA:13Þ

with RMSE2l FIGURE A2. A graphical illustration of inter-sensor and intrasensor clusters for No ¼ 4, where every number n represents a measurement received by sensor n at different times. Inter-sensor clusters are represented by ellipses, whereas intra-sensor clusters are by lines.

( ) Kt X 1X l 2 ¼ kxk;t x^ k;t k ; Kt k¼1 t[T

ðA:14Þ

k

where xˆlt is a posterior mean estimate of xt for lth run, and Tk ¼ fmax(t0k , ˆt0k ), min(t1k , t1k )g, which have been defined in MoP1.


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