Distributed multi-target tracking using joint probabilistic data

Ann. Telecommun. (2011) 66:553–566 DOI 10.1007/s12243-010-0224-9

Distributed multi-target tracking using joint probabilistic data association and average consensus filter Tohid Yousefi Rezaii & Mohammad-Ali Tinati

Received: 2 February 2010 / Accepted: 29 October 2010 / Published online: 20 November 2010 # Institut Télécom and Springer-Verlag 2010

Abstract In this paper, a distributed multi-target tracking (MTT) algorithm suitable for implementation in wireless sensor networks is proposed. For this purpose, the Monte Carlo (MC) implementation of joint probabilistic dataassociation filter (JPDAF) is applied to the well-known problem of multi-target tracking in a cluttered area. Also, to make the tracking algorithm scalable and usable for sensor networks of many nodes, the distributed expectation maximization algorithm is exploited via the average consensus filter, in order to diffuse the nodes’ information over the whole network. The proposed tracking system is robust and capable of modeling any state space with nonlinear and non-Gaussian models for target dynamics and measurement likelihood, since it uses the particlefiltering methods to extract samples from the desired distributions. To encounter the data-association problem that arises due to the unlabeled measurements in the presence of clutter, the well-known JPDAF algorithm is used. Furthermore, some simplifications and modifications are made to MC–JPDAF algorithm in order to reduce the computation complexity of the tracking system and make it suitable for low-energy sensor networks. Finally, the simulations of tracking tasks for a sample network are given. Keywords Target tracking . Monte Carlo simulations . JPDAF . Expectation Maximization . Consensus filter T. Yousefi Rezaii (*) : M.-A. Tinati Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran e-mail: [email protected] M.-A. Tinati e-mail: [email protected]

1 Introduction Current technology has enabled the development of sensor networks, distributed ad hoc networks of hundreds or thousands of nodes, each capable of sensing, processing, and communication. The ability to track targets is essential in many areas, such as military applications including missile defense and battlefield situational awareness, civilian applications are also ever-growing, ranging from traditional applications such as air traffic control and building surveillance to emerging applications like supply chain management and wildlife tracking. So far, much of the theory of tracking was developed for centralized processing of data from a relatively small number of radars or similar large devices endowed with plenty of power and high-bandwidth communications. In this paper we develop the multi-target tracking (MTT) problem in distributed manner in order to simultaneously make the tracking system scalable and usable for large networks while providing better tracking performance by using the spatial diversity obtained by spreading the sensors in the desired area to be covered. The MTT problem is not a trivial extension of singletarget tracking but rather a challenging topic of research. In MTT scenarios, there is a combinatorial explosion in the space of possible multiple target trajectories due to the uncertainty in the association of observed measurements with known targets in each step. This problem arises due to the fact that in most practical tracking applications, the sensors yield unlabeled measurements of targets. Theoretically, the standard recursive Bayesian filtering techniques can be applied directly to the joint state space of the targets, but computing the filtered distribution over the multi-target state and dealing with the combinatorial explosion of possible states due to the data-association ambiguity is

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difficult in practice. So, the main challenge in tracking multiple targets is to manage the computational complexity of the problem while still providing reasonable tracking performance [1]. Two main approaches are developed so far, to solve the data-association problem. The multiple hypotheses tracker (MHT) [2] which attempts to keep track of all possible association hypotheses over time. An advantage of this approach is that the number of tracks need not be known a priori since track initiations and terminations are explicitly hypothesized. However, MHT approach, in spite of its desired capabilities, suffers from large storage space and exponentially increasing processing. The alternative approach is the joint probabilistic data-association filter (JPDAF) [3–5] in which during each time step, the infeasible hypotheses are pruned away using a gating procedure in order to reduce the computational complexity. For this reason, the JPDAF approach is used, however in variational form, to solve the data-association problem in the proposed system and make it more suitable for online implementation. The main shortcoming of the JPDAF is that, to maintain the tractability, the final estimate collapses to a single Gaussian, thus discarding pertinent information. The linear and Gaussian models assumptions are often made by some authors to simplify hypotheses evaluation for target originated measurements, so implementation by Kalman filter is sufficient in this sense. One alternative is to use the extended Kalman filter instead [6]. However, the performance of such an algorithm degrades as the nonlinearities in models become more severe. More recently, strategies have been proposed to combine the JPDAF with particle techniques to accommodate general nonlinear and non-Gaussian models [7–10]. The Monte Carlo implementation of JPDAF, MC–JPDAF, developed in [10], can be considered as the first comprehensive algorithm that uses the Monte Carlo methods to solve the data-association problem while efficiently taking into account the dataassociation problem. So, the proposed method can be used for the general case of nonlinear and non-Gaussian dynamics and measurement likelihood models. In order to design a distributed MTT algorithm, it is necessary to map an MTT solution onto a sensor network platform with diverse resource limitations, including power, sensing, communication, and computation because data collection, processing, and dissemination all come at the cost of resource expenditure. Currently, several distributed target tracking algorithms have been developed [11–15]. In [11], the distributed particle filter is based on factorizing the likelihood and forming parametric approximations to products of likelihood factors (using the particles and their associated likelihoods as training data). The model parameters are then exchanged between sensor nodes instead of the data or exact particle information. In the algorithm

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proposed in [12], the distributed particle filters run on a set of uncorrelated sensor cliques that are dynamically organized based on moving target trajectories. Two distributive computing algorithms are developed in which the estimated Gaussian mixture model (GMM) parameters are transmitted instead of the whole particles. Using a GMM to approximate the posterior distribution is also adopted in [13], where the estimated parameters of GMM are transmitted to a fusion center. However, failure of the centralized unit is vital to the entire network. In [14], also a centralized unit is used, which receives the calculated local statistics of particles from network nodes. In this method, the particles are distributed in the sensor network. In [15, 16], the distributed expectation maximization (EM) algorithm is developed and used to approximate the global GMM parameters for the posterior distribution of weighted particles. In this approach, the global sufficient statistics for a given node is approximated using the local summary parameters of that node and the global summary parameters of the neighbor nodes. However, this method is only designed for single-target tracking situations, where the algorithm does not encounter with data-association problem which is common in multi-target scenarios. In this paper, we have developed a distributed multitarget tracking algorithm using the simplified version of MC–JPDAF [10] in order to solve the problem of MTT for the general case of nonlinear and non-Gaussian models for target dynamics and measurement likelihood, in the presence of false alarms, which we call it sequential Monte Carlo (SMC)–JPDAF. In the proposed method, each sensor estimates the marginal posterior densities of the individual targets through the Monte Carlo simulations and then calculates the local GMM parameters for these distributions. In continuation, each node obtains the global GMM parameters for the considered distributions by using the average consensus-filtering technique. This iterative procedure is repeated for consecutive time steps where, at each time step, the global approximations for the individual posterior densities are obtained from each node. The remainder of this paper is organized as follows: Section 2 discusses the general MTT scenario in sequential Bayesian framework, including models for target dynamics, data association, and measurement likelihood. Our proposed MTT system and the block diagram of the algorithm are presented in Section 3. In this section, the SMC–JPDAF framework, formulation, and the simplifications made, are discussed. Also, formulation of distributed EM algorithm for Gaussian mixtures of multiple targets used in proposed MTT system is described. The simulation results for a given network are given in Section 4. The results of the proposed system are compared to those of the several centralized algorithms. Finally, the conclusion remarks are included in Section 5.

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2 General MTT scenario The main aim of general Bayesian sequential tracking is to recursively estimate the filtering distribution pðxt jz1:t Þ, where xt is the joint state vector of the multiple targets and z1:t ¼ ðz1 :::zt Þdenotes all the observations up to the current time step t. It is assumed that the initial density pðx0 jz0 Þ of the state vector is available. Then, the posterior density may be obtained recursively, in two stages called prediction and update as follows [17]: Z pðxt jz1:t1 Þ ¼ pðxt jxt1 Þpðxt1 jz1:t1 Þdxt1 ð1Þ

In what follows, several models used in MTT scenarios are described. The evolution of the joint state space of the K slowly maneuvering targets in the xy plane is considered to be of the form described in [5] (the nearconstant velocity model). Also the state evolution of each target is assumed to be independent of the others. So the state of the kth target in the xy plane comprises its position and velocity as:   ð5Þ xk;t ¼ xk;t ; x k;t ; yk;t ; y k;t So the matrix form of the state transition equation is: xk;t ¼ Axk;t1 þ vk;t

pðzt jxt Þpðxt jz1:t1 Þ pðxt jz1:t Þ ¼ pðzt jz1:t1 Þ

ð2Þ

where the prediction step as in Eq. 1 follows from marginalization, and the new updated distribution as in Eq. 2 is obtained through a direct application of Bayes’ rule. The above recursions require the specification of a probabilistic model for the state evolution pðxt jxt1 Þ, known as dynamic function or system equation, and the measurement likelihood pðxt jz1:t Þ, also called the measurement equation. This recursive propagation of the posterior density is only a conceptual solution in general and cannot be determined analytically. The tracking recursion yields closed-form expressions in only a small number of cases. The Kalman filtering [18] for linear and Gaussian dynamic and likelihood models are one of such cases. For the general nonlinear and non-Gaussian models, the SMC methods, also known as particle filters or condensation, have gained a lot of popularity in recent years as the numerical approximation strategy to compute the tracking recursion. In general particle-filtering approach, the main idea is to sample and update then new particles with corresponding oN ðnÞ Importance weights as wðnÞ ; x , given the weighted set t t n oN n¼1 ðnÞ ðnÞ of samples of the previous time step, wt1 ; xt1 and the n¼1 well-defined   distribution called the proposal distribution ðnÞ q xt jxt1 ; zt . The new particles are generated from the proposal density, which may depend on the previous particles set and the current observations, that is:   ðnÞ ðnÞ xt  q xt jxt1 ; zt ; n ¼ 1:::N ð3Þ

where T is the sampling period, A is the state transition matrix, and vk;t is the process noise for the kth target and assumed to be zero mean Gaussian distributed with known covariance matrix defined as: 2

1 T 60 1 6 A¼4 0 0 0 0

ðnÞ

ðnÞ

/ wt1

ðnÞ

ðnÞ

ðnÞ

pðzt jxt Þpðxt jxt1 Þ

ð4Þ

ðnÞ ðnÞ qðxt jxt1 ; zt Þ

n o ðnÞ ðnÞ N wt ; xt

So new particle set, distributed according to pðxt jz1:t Þ.

n¼1

is approximately

s 2k;x T 2 =2 s 2k;x T 0 0

0 0 s 2k;y T 3 =3 s 2k;y T 2 =2

3 0 7 0 7 7 s 2k;y T 2 =2 5 2 s k;y T

So, we have the individual target dynamics distributions. The measurements are assumed to be available from No observer sensors at locations Poi ; i ¼ 1; :::; No . So the joint measurement vector from No observers at each time interval is:     z ¼ z1 :::zNo ; where zi ¼ zi1 :::ziM i ð8Þ M i is the total number of measurements in the ith observer, comprising Mci clutter measurements and MTi measurements arising from the targets to be tracked. In general, Mci and MTi varies from an observer to another one and also in different time instances in the same observer. To deal with the data-association problem, we consider the association variables presented in [10]. The measureto target association hypothesis is defined as l ¼ ment   l1 :::lNo where li ¼ ri ; MCi ; MTi is the measurement to target association hypothesis for the measurements at the ith observer. The elements of the association vector ri are given as: 

wt

2 2 3 3 s k;x T =3 0 0 6 s 2 T 2 =2 0 07 k;x 7Σ k ¼ 6 6 1 T5 0 4 0 1 0

ð7Þ

rji ¼

The new particles’ weights are defined as follows:

ð6Þ

0 if measurement j at observer i is due to clutter k 2 f1:::Kg if measurement j at observer i stems from target k

ð9Þ The target to measurement association hypothesis e l¼ 1 eNo e ðl :::l Þ is defined in a similar fashion, where e li ¼ ðeri ; MCi ; MTi Þ is the target to measurement association

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hypothesis for the measurements at the ith observer. The elements of the association vector eri are obtained as follows:  erki ¼

0 if target k is undetected at observer i to clutter j 2 f1:::M i g if target k generated measurement j at observer i

ð10Þ The two sets of above hypotheses are equivalent. Considering that the observer sensor i measures the range Rij and the bearing qij between itself and a target, the individual measurements at the ith observer are zij ¼ ðRij ; qij Þ; j ¼ 1; :::; M i . If the range and bearing are assumed to be corrupted by independent Gaussian noise, the likelihood for the jth measurement, under the hypothesis that it is associated with the kth target becomes piT ðzij jxk Þ ¼ Nðzij jbzij ; Σ iz Þ

ð11Þ

where Σ iz ¼ diagðs 2Ri ; s 2qi Þ is the fixed and known diagonal covariance with the individual noise variances s 2Ri and s 2qi , for i bi ; b the range and bearing, respectively. bzij ¼ ðR j q j Þ is the mean vector of the Gaussian distribution in Eq. 12 and is given by:     1=2 b i ¼ xk  xi 2 þ yk  yi 2 R ð12Þ k o o   i i 1 yk  yo b qk ¼ tan xk  xio

ð13Þ

where Pio ¼ ðxio ; yio Þ is the position of the ith observer. In order to estimate the unknown association hypothesis within a Bayesian framework, definition of the prior distributions for these hypotheses is necessary. As the prior distributions, we follow those described in [10] and [5]. The prior for the association hypothesis is assumed to be independent of the state and past values of the association hypothesis. For the measurement to target association hypothesis, it is assumed that the prior factorizes over the observers, i.e.: pðlÞ ¼

No Y

pðl Þ i

ð14Þ

i¼1

And so for each of the observers, the prior is assumed to be: pðli Þ ¼ pðri jMCi ; MTi ÞpðMCi ÞpðMTi Þ

ð15Þ



1 pðri jMCi ; MTi Þ ¼ Nli ðMCi ; MTi Þ

ð16Þ

i

pðMCi Þ ¼ ðliC ÞMC expðliC Þ=MCi !  pðMTi Þ ¼

K MTi

 i Mi PD T ð1  PD ÞKMT

ð17Þ

ð18Þ

PD is the detection probability of each target in each observer. As it is clear from Eqs. 17, 18, to 19, prior distribution for the association vector is assumed to be uniformly distributed over all valid hypotheses, number of clutter measurements are assumed to follow a Poisson distribution with rate parameter liC and prior distribution for the target measurements is assumed to follow a binomial distribution. The prior for the target to measurement association hypothesis follows the same structure. Furthermore, it is possible to obtain a factorization form for target to measurement association hypothesis prior as: i pðe l Þ ¼ pðMCi Þ

K Q

pðerik jeri1:k1 Þ k¼1 8 < 1  PD if j ¼ 0 i pðerik ¼ jjeri1:k1 Þ / 0 if j > 0 and j 2 fer1i :::erk1 g : PD otherwise   where Mki ¼ M i  fl : er il 6¼ 0; l ¼ 1:::k  1g

ð19Þ

3 The proposed distributed MTT system In this section, several parts of the proposed distributed multi-target tracking algorithm will be introduced. In what follows, the SMC–JPDAF algorithm, which is used for tracking and data-association purposes, is presented. In continuation, the distributed GMM algorithm using consensus filtering will be formulated for multi-target scenarios, which is used to spread the tracking task among the network nodes. Finally, the block diagram of the proposed MTT algorithm is described. 3.1 SMC–JPDAF framework As mentioned previously, JPDAF approach due to its simplicity and low computational complexity in contrast with MHT approach is the most widely applied method in MTT problems considering the data-association uncertainty. For this reason, we have chosen this approach to solve the data-association problem in this paper. Several implementation strategies for JPDAF method have been proposed in literature depending on what application area used. Since the target dynamics and measurement likelihood models in target tracking applications are nonlinear and non-Gaussian in general, the selected Bayesian framework should have the ability to estimate and track such nonlinear and non-Gaussian models. For this reason, the Monte Carlo implementation of JPDAF, but in simplified form, is used in our distributed MTT algorithm. Due to the particle-filtering methods used in MC– JPDAF, it has the ability to track the arbitrary proposal distributions.

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The main idea of JPDAF is to recursively update the marginal filtering distributions for each of the targets   pk xk;t jz1;t1 ; k ¼ 1; :::; K, through the Bayesian sequential estimation recursion Z       pk xk;t jz1;t1 ¼ pk xk;t jxk;t1 pk xk;t1 jz1;t1 dxk;t1

imately distributed according the marginal filtering distribution at the previous time step pk ðxk;t1 jz1:t1 Þ. At the current time step, the new samples for the target state are generated from a suitably constructed proposal distribution, i.e.:

ð20Þ

In our distributed MTT system, in order to reduce the computational complexity of the proposed tracking algorithm, we have used the prior distribution as the proposal distribution for kth target rather than complicated ones proposed in literature, for example in [10]. This will significantly reduce the execution time of the tracking algorithm, while still maintaining the tracking performance, as discussed later in the Section 4. So the proposal distribution will be of the form:

Due to the data-association uncertainty, the filtering step cannot be performed independently. In JPDAF, the likelihood for the kth target is assumed to be: " # No Mi Y X i i i i pk ðzt jxk;t Þ ¼ bjk þ bjk pT ðzj;t jxk;t Þ ð21Þ i¼1

j¼1

i where bijk ¼ pðerk;t ¼ jjz1:t Þ; j ¼ 1:::M i , is the posterior probability that shows how the kth target is associated with the jth measurement in the ith observer, and bi0k is the posterior probability that the kth target is undetected. Furthermore, the likelihood is assumed to be independent over the observers. With the definition of the likelihood as in Eq. 21, the filtering step is as follows:

pk ðxk;t jz1:t Þ / pk ðzt jxk;t Þpk ðxk;t jz1:t1 Þ

ð22Þ

All that remains is to compute the posterior probabilities of the marginal associations bijk as: X i i ¼ jjz1:t Þ ¼ pðe lt jz1:t Þ ð23Þ bijk ¼ pðerk;t i i i e e fl t 2Λt :e rk;t ¼jg e i is the set of all joint target to measurement where Λ t association hypotheses for the data at the ith observer. The posterior probability for the joint association hypothesis i pðe l jz1:t Þ can be expressed as [10]: i Y i i pðe lt jz1:t Þ / pðe lt ÞðV i ÞMC prj;ti ðzij;t jz1:t1 Þ ð24Þ j2Ii i where the expression pðe lt Þ is given from Eq. 19, V i is the volume of the measurement space for the ith observer defined as V i ¼ 2pRimax , where is Rimax the maximum range for the ith observer, and Ii ¼ fj 2 f1:::M i g : rji 6¼ 0g. The expression prj;ti ðzij;t jz1:t1 Þ in Eq. 24 is the predictive likelihood for the jth measurement at the ith observer using the information from the kth target, given in the standard form by: Z pk ðzij;t jz1:t1 Þ ¼ piT ðzij;t jxk;t Þpk ðxk;t jz1:t1 Þdxk;t ð25Þ

In Monte Carlo framework, the predictive likelihood in Eq. 25 is approximated using the Monte Carlo samples from the proposal distribution. It is assumed that for the kth target, ðnÞ ðnÞ the set of samples fwk;t1 ; xk;t1 gNn¼1 are available, approx-

ðnÞ

ðnÞ

xk;t  qk ðxk;t jxk;t1 ; zt Þ; n ¼ 1:::N

ðnÞ

ðnÞ

ð26Þ

ðnÞ

qk ðxk;t jxk;t1 ; zt Þ ¼ pk ðxk;t jxk;t1 Þ

ð27Þ

Using these Monte Carlo samples, the predictive likelihood in Eq. 25 can be approximated as: pk ðzij;t jz1:t1 Þ 

N X

ðnÞ

ðnÞ

ak;t piT ðzij;t jxk;t Þ

ð28Þ

n¼1

where, using the definition in Eq. 27, the predictive weights ðnÞ ðnÞ are equal to importance weights, ak;t ¼ wk;t ; n ¼ 1; :::; N . So the update equation for the importance weights is as: ðnÞ

ðnÞ

ak;t / wk;t1

ðnÞ

ðnÞ

pk ðxk;t jxk;t1 Þ ðnÞ

ðnÞ

qk ðxk;t jxk;t1 ; zt Þ

N X

;

ðnÞ

ak;t ¼ 1

ð29Þ

n¼1

The approximation to the predictive likelihood can now be substituted into Eq. 24 to obtain the estimate of the joint association posterior probabilities, from which approximations for the marginal target to measurement association posterior probabilities can be computed according to Eq. 23. These approximations can be used in Eq. 22 to estimate the target likelihood. Finally, setting the new importance weights to: ðnÞ

ðnÞ

N X

ðnÞ

wk;t ¼/ wk;t1 pk ðzt jxk;t Þ;

ðnÞ

wk;t ¼ 1

ð30Þ

n¼1 ðnÞ

ðnÞ

leads to the sample set fwk;t ; xk;t gNn¼1 being approximately distributed according to the marginal filtering distribution at the current time step pk ðxk;t jz1:t Þ. 3.2 Formulation of distributed EM algorithm for Gaussian mixtures of multiple targets used in proposed MTT system In this section, we will not be using the time index t anymore for simplicity. The obtained equations can be used for the consecutive time steps. The expectation

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maximization algorithm for Gaussian mixtures in distributed manner deals with calculation of the local summary statistics in all observers for each target and then using these local summary statistics and the neighbor sensors global summary statistics of the previous time to estimate the global summary statistic for the next time step. For this reason, a sensor network with K targets to be tracked, J independent observers, M j measurements at the jth observer and N particles as the number of samples to be generated for each target in each time step, is assumed. The desired posterior distribution for each target is assumed to be a Gaussian mixture with C mixture probabilities ak;j;c ; ðk ¼ 1; :::; K; j ¼ 1::; J ; c ¼ 1; :::; CÞ. The Gaussian ðnÞ mixture distribution for state xk;j is: ðnÞ

pðxk;j jy k Þ ¼

C X

ðnÞ

ak;j;c pðxk;j jmk;c Σ k;c Þ

ð31Þ

c¼1

where y k is the set of the distribution parameters to be estimated for target k, where y k ¼ fak;j;c ; mk;c ; Σ k;c g. ðnÞ The probability for the state xk;j in cth mixture follows a Gaussian distribution with mean mk;c and variance Σ k;c : T ðnÞ ðnÞ 1 ðnÞ  1 ðx m Þ Σ 1 ðx m Þ pðxk;j jmk;c ; Σ k;c Þ ¼ pffiffiffiffiffi 1=2 e 2 k;j k;c k;c k;j k;c 2pΣ k;c 

ð32Þ The local summary quantities for the kth target at the jth observer will become: aqk;j;c ¼

bqk;j;c ¼

g qk;j;c

1 N

N X

aqk;j;c;n

ð33Þ

n¼1

N 1 X ðnÞ aq x N n¼1 k;j;c;n k;j

N T 1 X ðnÞ ðnÞ ¼ aq ðx  mqk;j;c Þðxk;j  mqk;j;c Þ N n¼1 k;j;c;n k;j

ð34Þ

ð35Þ

where the index q in Eqs. 33, 34, and 35 denotes the iteration number for the distributed GMM algorithm and mqk;j;c is the local mean in the jth observer for the cth mixture component. The expression for the posterior probability aqk;j;c;n is obtained from the set of distribution parameters in the previous time step y q1 as follows: k ðnÞ

aqk;j;c;n ¼

q1 q1 aq1 k;j;c pðxk;j jmk;c ; Σ k;c Þ C P ðnÞ q1 q1 aq1 k;j;l pðxk;j jmk;l ; Σ k;l Þ

l¼1

ð36Þ

The calculation stage of the local summary quantities is called expectation step (E-step). In continuation, the global summary quantities for each sensor can be obtained using the local summary quantities of that sensor and the global summary quantities of its neighboring sensors. We use uqk;j;c ¼ ½N aqk;j;c ; bqk;j;c ; g qk;j;c T to denote the local summary quantities and # qk;j;c ¼ ½# qk;j;c ð1Þ; # qk;j;c ð2Þ; # qk;j;c ð3ÞT to denote the global summary quantities in node j. Now, the discrete form of the consensus filter can be used to obtain approximations for the global summary quantities in each node (consensus step) as follows: 2 3 X # qþ1 ¼ # q þ d j 4 ð# q  # q Þ þ ðuq  # q Þ5 k;j;c

k;j;c

k;m;c

k;j;c

k;j;c

k;j;c

m2Nj

ð37Þ where δj is the step size parameter for the jth observer, determining the updating rate and should satisfy the following condition, in order to have a stable consensus filter. dj 

1 degmax

ð38Þ

where degmax is the maximum degree of nodes for a specific network. In the maximization step (M-step), the global distribution parameter set is obtained at each observer by using the following equations: mqþ1 k;c ¼

# qþ1 k;j;c ð2Þ N # qþ1 k;j;c ð1Þ

;

Σ qþ1 k;c ¼

# qþ1 k;j;c ð3Þ N # qþ1 k;j;c ð1Þ

ð39Þ

The block diagram of the distributed EM algorithm for GMM parameters of multiple targets in a single observer is depicted in Fig. 1. 3.3 Block diagram and pseudo-code of the proposed distributed MTT algorithm In this section, we present and discuss the block diagram of our proposed distributed MTT algorithm. Starting with the initial marginal distributions for target at each sensor, the particles at current time step are generated from the proposal distribution, as of Eq. 27. Then, the distributed EM algorithm is performed for sufficient iterations at each node and the global distribution set at the current time step is calculate and saved for each observer. In continuation, the new particle set is generated from the calculated GMM distributions for each target. The SMC–JPDAF algorithm with gating procedure is executed in the next step at each network node to obtain the new sample set with the corresponding weights for each target and it is approximately distributed according to the marginal posterior

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559

Fig. 1 Block diagram of the distributed EM algorithm for GMM parameters of multiple targets in jth observer

distribution of the corresponding target. This procedure precedes recursively for the consecutive time steps. The block diagram of the proposed MTT system for a single time step is shown in Fig. 2. According to this figure, starting with the initial distributions for targets’ states, the marginal distribution for each target is sampled in each observer. Then the distributed GMM algorithm runs for q iterations. The advantages of the proposed distributed MTT algorithm may be discussed form two major aspects. First, the main advantage of the proposed distributed MTT algorithm is that, the approximations for the marginal posterior distributions of interest are available at each of the sensor nodes, i.e., the tracking results of networks’ nodes converge to each other by means of the consensus filter used. Furthermore, tracking performance would improve by increasing the number of sensor nodes. This is due to the spatial diversity, obtained by spreading the sensors in the area of interest. So, the proposed algorithm is robust to the failure of some nodes, i.e., the algorithm has the ability to track the multiple targets, even with a few number of the sensor nodes. This feature is desired in military applications, where the high-tracking performance and robustness against nodes failure are of interest. As the second aspect, the advantage of the proposed algorithm is that, it has the ability to use sensor nodes with low processing ability as the network nodes and still maintain the desired tracking performance. In centralized implementation of MTT, large number of particles is needed to reach a desired tracking performance, leading to have sensor nodes with higher processing ability and so higher energy consumption. However, as it is shown in the next section, the proposed distributed MTT algorithm could reach the same tracking performance by rather a fewer number of particles for each observer. In other words, the tracking task will distribute in whole networks nodes. This is the case of interest in implementation of low-cost sensor networks with relatively high-tracking ability and reliability.

4 Simulation results This section includes the simulation results and finally the comparison results of different multi-target tracking algorithms with the proposed one. First, several assumptions, parameters, and models considered for the simulations are introduced. In what follows, all location positions and distance measures are in meters, angles in radians, times in seconds, and velocity measures in meters per second. Also, the near-constant velocity model in [5] is used with standard deviation for system noises as s x ¼ s y ¼ 0:1 and for measurements noises s R ¼ 5 and s q ¼ 0:05. The discretization time T is set to 1 s. All the simulations are performed for two target case, i.e., K ¼ 2. So the initial state vectors for two targets are assumed to be ½50 1 50 1:5 and ½50 1 0  0:5. For the sake of creating the initial conditions for the sequential Monte Carlo algorithm, each of the targets’ state is sampled following the Gaussian distribution with variance 5 and 0.5 for location and velocity, respectively. The detection probability for each of the targets at each observer is assumed to be the same and is set to pD ¼ 0:8. Also, the rate parameter for the Poisson distribution of the clutter measurements is assumed to be lC ¼ 0:5. All the sensors are assumed to be of the same detection and communication ability, and the maximum range detection distance is Rmax ¼ 100. Considering the above assumptions the covariance matrix for system noise in Eq. 7 is calculated as: 2

0:0033 6 0:005 Σk ¼ 6 4 0 0

0:005 0 0:01 0 0 0:0033 0 0:005

3 0 0 7 7 0:005 5 0:01

ð40Þ

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Fig. 2 Block diagram of the proposed distributed multi-target tracking algorithm

Also, the covariance matrix for the range and bearing observations in Eq. 11 is:

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The initial values for the importance weights are set to 1/N, where N is the number of particles. The results for each of the algorithms are obtained for the particle numbers of 10, 25, 50, 100, 200, 300, 500, and 700. The estimated track for each for the algorithms is plotted for the number of particles equal to 100. For comparison, the root mean squared error (RMSE) curves and the time of execution plots versus the number of particles are computed. The RMSE for each algorithm is obtained by executing the algorithm L times and computing the second root of the mean squared error of each repetition. In all of the experiments, the number of repetitions is set to L=20.

The simulations and comparisons are conducted in two major phases. The first phase deals with three different centralized tracking algorithms. As the first algorithm, the SMC–JPDAF algorithm followed by the gating procedure is simulated for two dependent observers (algorithm 1). The expression “two dependent observers” means that the observations of two observers are sent to a central node and the tracking algorithm uses both of these two observations as the observation data. In this case, one of the two sensor nodes is arbitrarily considered as the central node. The true track of two targets (solid lines) and the estimated track (colored dashed lines) by using algorithm 1 is plotted in Fig. 3. In this figure, the locations of two observers are plotted as magenta circles and the number of particles is set to 100. Also, the RMSE curve versus the number of particles for algorithm 1 is plotted in Fig. 4 and the time of execution for a single iteration of the program in central node, versus the number of particles, is plotted in Fig. 5.

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As it is obvious from the above mentioned figures, the RMSE monotonically decreases and so the algorithm performance increases as the number of particles increases. Furthermore, the time of execution increases linearly as the number of particles increases. This property is an advantage for a tracking algorithm considering that many of tracking algorithms have an exponential curve for the time of execution versus the number of particles. In the second centralized algorithm (algorithm 2), the SMC–JPDAF algorithm with gating procedure is simulated for five dependent observers. Also in this case, one of the sensors is arbitrarily selected as the central node. The centralized network topology is depicted in Fig. 6 while the estimated track, RMSE, and timing curves are plotted in Figs. 7, 8, and 9, respectively. In Fig. 6, the communication link between nodes is depicted by red solid lines. As it was expected, the tracking performance is increased in this case, since it applies the observation data of five observers rather than two. Instead, the time of execution has increased significantly since the computational cost of the algorithm has increased. Also, the power consumption of this kind of network is relatively high, since it needs the transmission of all observation data from all nodes to the central node, resulting a short life time for the whole network. Furthermore, the tracking system is

highly dependent on the central node and the tracking will stop with any failure of the central node. In the third centralized tracking algorithm (algorithm 3), the SMC–JPDAF algorithm followed by gating procedure is simulated with five independent observers, i.e., in each of the observers a separate SMC–JPDAF algorithm is running without any communication between nodes. So in each observer, the target will be tracked by means of only the observation data obtained by that observer. In this case, each of the sensor nodes may be considered as the central node. So the tracking algorithm in each observer will be highly sensitive to the measurement errors, environment obstructions, and node failures. Furthermore, due to the independent observations used in each observer, the estimated track and so the tracking performance of nodes will highly vary from one to the other. The network topology of the algorithm 3 is shown in Fig. 10a. The dashed blue lines indicate the loss of communication link between network nodes. As it was mentioned before, there was no communication link between nodes. Figure 10b to f show the true and estimate tracks of algorithm 2 for each of the five observers.

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Also the RMSE plot for each of the observers is shown in Fig. 11. As it is obvious, the tracking performance of nodes is not the same and the main shortcoming of this algorithm is how to find the node with minimum RMSE in each time step and extract the tracking results from that node. As it is shown in Fig. 12, the execution time of this algorithm is also nearly linear and as it was expected, since in algorithm 3 each node only uses the observation data of one sensor, the execution time for algorithm 3 is less than that of algorithms 1 and 2. As the second phase of the simulation results, the proposed distributed multi-target tracking algorithm is simulated. For simplicity, a sensor network with only five nodes is considered, as depicted in Fig. 13a considering the maximum detection range of 100 m for each of the observers, the adjacency matrix A for this network is as follows 3 2 0 1 0 0 1 61 0 1 0 07 7 6 7 ð42Þ A¼6 60 1 0 1 07 40 0 1 0 15 1 0 0 1 0

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node j. The adjacency matrix above leads to a connected network as shown in Fig. 13a; in this figure, the solid red lines imply the existence of transmission link and the dashed blue lines imply the loss of transmission link between nodes. In the proposed algorithm, the number of GMM mixtures, C is set to 3. The number of iterations of the distributed GMM algorithm in each time step depends on the desired performance for the algorithm. For our simulations this value is set to 10. Increasing the number of iterations will lead to performance improvement, increasing the distributive property of the algorithm and more important, further converging the performance and inference ability of network nodes. In the other hand, increasing the number of iterations for GMM algorithm will increase the execution time of the algorithm. So, depending on the application, a kind of trade off should be hold between the tracking performance and execution time. The step size parameter for the consensus filter is empirically set to 0.005. The true tracks and estimated tracks of the targets using the proposed algorithm are depicted in Fig. 13b–f. The results are plotted for each of the five sensor nodes and the number of particles is set to 100. Also the RMSE plots for sensor nodes are shown in Fig. 14. As it is obvious from Figs. 13 and 14, the tracking performance of the proposed algorithm for the given number of particles is better than the three previous centralized algorithms. Also, it is seen that the estimated track and tracking performance of the network nodes are nearly the same. So the tracking results are available from each of nodes and this is a great advantage for a target tracking network. The tracking performance of the proposed algorithm is less dependent of node failures, i.e., one node failure only means reduction of one of adjacent nodes for the neighbors of that node in whole network and this will negligibly reduce the tracking performance of the network especially for larger networks. Further, comparison of the results will be carried out later in this section. Figure 15 shows that the execution time of the algorithm 4 also have a linear relation with the number of particles.

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1. For a given number of particles the proposed algorithm obtains the best tracking performance with the least RMSE value. 2. The proposed algorithm can reach a specified value for RMSE by consuming the least time compared to the other centralized algorithms. 3. The proposed distributed tracking algorithm is less sensitive to the node failures and the tracking results are available from each of the network nodes.

5 Conclusion remarks In this paper, a kind of distributed multi-target tracking algorithm is proposed that could be used in arbitrary large 25

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and dense wireless sensor networks. The tracking algorithm has the ability to track any general state space model with nonlinear functions and non-Gaussian distributions for system and measurement models. In order to jointly solve the tracking and data-association problems, the simplified version of the Monte Carlo implementation of well-known JPDAF algorithm is used in network nodes. Also to make distributive inferences in whole network, the distributed implementation of GMM algorithm is applied. In this manner, the consensus filter as a low-pass filter is used to make global inferences from the local inferences of the network nodes. So the proposed distributed multi-target tracking algorithm is robust against various measurement errors and node failures which are common in sensor networks. Also, the proposed algorithm has the ability to track multiple targets in cluttered areas, due to unwanted targets that make unwanted observations in sensor nodes. These claims are proved by simulation results where comparisons are made by three different centralized algorithms under the same conditions. At last, the performance improvement of tracking precision and time of execution for the proposed algorithm are proven. Finally, the implementation of the above proposed system in a real sensor network platform will be considered as the future work, where the above claimed results and performance will be proved practically. To address this, we intend to use a kind of real test bed, like web-based MoteLab, developed and deployed at Harvard University. MoteLab consists of a set of permanently deployed sensor network nodes connected to a central server which handles reprogramming and data logging while providing a web interface for creating and scheduling jobs on the test bed. Each node for example consists of a TI MSP430 processor running at 8 MHz, 10 KB of RAM, 1 Mbit of Flash memory and a Chipcon CC2420 radio operating at 2.4 GHz with a range of approximately 100 m. As the contribution of this work, it is expected that using a low-cost and lowenergy sensor network, where each node be capable of

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detecting just the range and bearing of the targets only within a restricted zone, it would be possible to reach the tracking accuracy of fully connected networks. Furthermore, each node is considered to have a limited processing resource. The network could be extended using the same sensors without replacing them with powerful ones. The other main characteristic of such a sensor network platform would be the robustness against node failures, i.e., the network would maintain the tracking task if some part of the network sensors stop working and more especially the tracking performance would not considerably degrade due to some node failures. It is also noteworthy that the network sensors would have significantly longer lifetime compared to the fully connected networks, particularly in large networks.

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12. Acknowledgment This work was partially supported by Iran Telecommunication Research Center. 13.

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