Multi-Target Tracking in a Two-Tier Hierarchical Architecture

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1 Multi-Target Tracing in a Two-Tier Hierarchical Architecture Jin Wei Department of Electrical Engineering University of Hawaii at Manoa Honolulu, HI 968, U.S.A. weijin@hawaii.edu Xudong Wang Department of Electrical Engineering University of Hawaii at Manoa Honolulu, HI 968, U.S.A. xudongw@hawaii.edu Vassilis L. Syrmos Department of Electrical Engineering University of Hawaii at Manoa Honolulu, HI 968, U.S.A. syrmos@hawaii.edu Abstract In this paper, a two-tier hierarchical architecture is proposed to address the multi-target tracing problem using a Particle Probability Hypothesis Density filtering algorithm. According to a proposed cluster scheduling method, the base station selects active clusters at each time step and determines their order for the sequential data fusion in the second level of hierarchy. Within each active cluster, sensors transmit their measurement-sets to the cluster head, which processes the information locally and estimates the number of targets and their states. The proposed architecture wors well even when the target dynamics and/or measurement process is severely nonlinear. The performance of this architecture is demonstrated in the application of bearing and signal strength tracing. Keywords: Particle Probability Hypothesis Density filter, Data Fusion, cluster scheduling, Gaussian Mixture Model. I. INTRODUCTION Multi-Target Tracing MTT has received many attentions these years. It involves joint estimation of unnown and time-varying number of targets and their states in cluttered environment [1,, 3]. Finite-Set Statistics FISST [5] uses Random Finite Sets RFSs to model the collections of multitarget states and measurements as set-valued entities, and provides a Bayesian framewor for MTT. As one of the RFS-based filters, Probability Hypothesis Density PHD filter [6] was developed as a recursion propagating the intensity function associated with the multi-target posterior. Two main types of PHD filters have been proposed. One is Particle PHD filter [7], which uses Sequential Monte Carlo techniques to approximate the posterior intensity function and uses clustering techniques to determine the states from the multi-modal empirical density. The main drawbac of this approach is the unreliable estimation of the target number. The other one, called the Gaussian Mixture PHD GMPHD filter [8], provides a closed-form solution to the PHD recursion, but is restricted to linear-gaussian target dynamics and measurement process. Extensions for the GMPHD filter, which use Extended or Unscented Kalman filters, allow for mildly non-linear dynamics. However, they still perform poorly when the nonlinearity is severe. In this paper, the Particle PHD filter is implemented and its reliability is improved by the proposed data fusion procedures. The simulation section will show that the proposed two-tier hierarchical architecture wors well even when the target dynamics and/or measurement process is severely non-linear. Furthermore, this architecture considers a more realistic model of the detection probability, whose value varies with the distance between the traced target and the sensor. In order to trade off computational complexity and estimation accuracy, an efficient cluster scheduling scheme is required for the proposed architecture. In 5, Mahler derived an objective function, called the Posterior Expected Number of Targets PENT, for multi-target sensor management based on the PHD filter [9, 1, 11]. Maximizing PENT has the effect of maximizing the number of detected targets and, to a lesser extent, the estimation accuracy of the target states [1]. In this paper, the proposed cluster scheduling algorithm is developed based on the PENT objective function, and the number of active clusters is determined dynamically in this algorithm. Since multiple clusters of sensors are active, data fusion procedures in both levels of hierarchy are required. In the context of RFS, some data fusion methods have been developed. For instance, a nearest-neighbor correlation method is proposed in [17]. This method is not practical since there are no rules to determine an optimal threshold value for the correlation. A sequential updating method is presented in [6, 13]. However, the authors did not mention how to determine the order of updating. [15] proposed a synchronized updating method. This method is also not appropriate for the proposed MTT algorithm in this paper, since it is derived based on Bayesian-RFS filter, not PHD filter. Furthermore, in [16], it mentioned that in PHD filtering algorithm a missed detection from one sensor can affect other sensors with more reliable detection. However, as far as we now, there are no methods to fix this problem until now. In this paper, the data fusion method is developed for each level of hierarchy. In the first level, the fusion method is based on an idea similar to the synchronized updating method in [15], but is implemented using Particle PHD filtering algorithm. This method can fix the problem mentioned in [16] by only fusing the higherquality data from sensors. In the second level, the estimates from the active Cluster Heads CHs are fused sequentially [6, 13] in a proposed fusion order. This method can also efficiently reduce the degeneracy problem in the Particle PHD filter through approximating the locally updated particles with a Gaussian Mixture Model GMM [14] and only transmitting 194

2 the parameters of the GMM between the active CHs. The rest of this paper is organized as follows. In section II, we introduce the RFS model for the MTT application and the Particle PHD filter. In section III, we introduce the PENT objective function and describe the proposed cluster scheduling method. Section IV describes the proposed data fusion methods. The simulation results in an application of bearing and signal strength tracing are presented in section V. Finally, some conclusions and potential extensions are discussed in section VI. II. PROBLEM FORMULATION A. Random Finite Set Model for Multi-Target Tracing The MTT problem can be modeled by RFS framewor [5]. Let χ be the single target state space, and M be the number of targets at time, then multi-target state at time is presented by X = x,1,x,,...,x,m } Ϝχ, where Ϝχ denotes the collection of all finite subsets of χ. Fora multi-target state X 1 at time step 1, each x 1 X 1 either continues to exist at time with probability p S, x 1, or dies with probability 1 p S, x 1. Therefore, for a given state x 1 X 1 at time 1, its behavior at the next time step can be modeled as the RFS S 1 x 1, that can tae on either x } when the target survives, or φ when it dies. A new target at time can arise either by spontaneous births which can be modeled by Γ, or by spawning from x 1 which can be modeled by B 1 x 1. Given a multi-target state X 1 at time 1, the multi-target state X at time is formed by union of the surviving targets and new targets, X = S 1 ζ B 1 ζ Γ, ζ X 1 ζ X 1 1 Similarly, let the measurement-set collected by the i th sensor at time to be Z [i] ϜZ[i]. A given target state x X is either detected with probability p D, x or missed with probability 1 p D, x. Consequently, the measurement from x at sensor i can be modeled by the RFS Θ [i] x, that can tae on either z } when the target is detected, or φ otherwise. The i th sensor can also receive a set C [i] of clutter. So, given a multi-target state X at time, the measurement-set collected by the i th sensor is formulated by Z [i] = [ x X Θ [i] x ] [i] C, Let Q be the number of sensors, then the RFS of measurements at time is modeled by [ ] Z = Z [1],Z[],...,Z[Q]. 3 The RFS X encapsulates all aspects of multi-target tracing problem, and Z encapsulates all sensor characteristics. The multi-target tracing can be posed as follows: given the measurement-set Z 1: collected from sensors up to time, the problem is to find ˆX that is the expectation of the posterior density function px Z 1:. B. Particle Probability Hypothesis Density Filter In the proposed architecture, Particle PHD filter is employed to estimate the number of targets and their states. The PHD filter is a computationally tractable alternative to the optimal multi-target Bayesian filter, and the Particle PHD filter is a SMC implementation of PHD filter. 1 Overview of Probability Hypothesis Density Filter: The PHD filter is an approximation developed to propagate the posterior intensity, a first-order statistical moment of the posterior multi-target state. The intensity υx is defined as follows. For a RFS X on χ with probability distribution P, the integral of υx over region S gives the expected number of elements of X that are in S, X N = S P dx = S υxdx, 4 It can be shown that the posterior intensity can be propagated in time via the PHD recursion. υ 1 x = + p S, ζf 1 x ζυ 1 ζdζ β 1 x ζυ 1 ζdζ + γ x, υ x = [1 p D, x]υ 1 x + p D,xg z xυ 1x κ z+ p D, ξg z ξυ 1 ξdξ z Z where f 1 ζ denotes the single target transition density, γ denotes the intensity of the spontaneous birth RFS, β 1 ζ denotes the intensity of the spawning birth RFS, p S, ζ denotes the probability that a target continues to exist given that its previous state is ζ, g x denotes the single target measurement lielihood, p D, x denotes the detection probability given a state x, and κ denotes the intensity of the clutter RFS. Note that κ can be modeled as r c, where r is the average number of clutter points per scan and c is the probability distribution of each clutter point. The local maxima of the intensity υ are points in χ with the highest local concentration of expected number of elements, and thus can be used to generate the estimates for the elements of X. Thus, we can estimate the states of targets by investigating the peas of PHD. Particle PHD Filtering Algorithm: The basic idea of Particle PHD filter is the propagation of a particle approximation to the posterior intensity function through the PHD recursion 5-6. A brief description is given in Table I [7]. For simplicity, we assume that the intensity of the spontaneous birth RFS Γ can be modeled as a Gaussian mixture of the form n γ, γ x = p i γ, N x; m i γ,,pi γ,

3 Table I PARTICLE PHD FILTERING ALGORITHM 1. Initialization Draw L particles from the prior PHD and assign weights: ω i = N, where N is the prior estimation of target number. L. At time 1 According to the proposal densities q x i 1,Z and p Z draw predicted particles: x i q x i 1,Z, i =1,...,L 1 ; p Z, i = L 1 +1,...,L 1 + J. and calculate the associated weights as: φ 1 x i,xi 1 w i 1 = q x i Z 1 J ω i 1,,...,L 1, γ x i p x i Z,i= L 1 +1,...,L 1 + J, where φ 1 x i,xi 1 = β 1 + p S, x i Update each [ of the weights ω i = where ψ,z 1 p D, x i x i x i xi 1 1 f 1 x i + ψ,z x i z Z κ z+c z = p D, x i C z = L 1 +J j=1 ψ,z g z x i x j w j 1. Calculate the total mass Ñ = L 1 +J j=1 ω j }, ω i L 1 resample, x i +J ω i Ñ to get } rescale the weights by Ñ to get ω i L,xi. ] xi 1. w i 1, } L,x i Ñ, where n γ,, p i γ,, mi γ,, P i γ,, i =1,...,n γ,, are given model parameters that determine the shape of the intensity. III. PROPOSED CLUSTER SCHEDULING METHOD The proposed scheduling method employs the PENT objective function to choose active clusters of sensors. A. PENT Objective Function Mahler introduced an objective function, called Posterior Expected Number of Targets PENT, for multi-target sensor assignment in [9, 1, 11]. The basic idea of this objective function is outlined as follows. Let x =x 1,...,x s be the future joint sensor state, then an objective function Ox,Z for sensor management can be constructed from the data-updated intensity υ x Z.The objective function mathematically describes what goals we want sensor management to maximize. Equation 4 provides the following choice for objective function: Nx,Z = υ x, x Z dx. 8 S Since the sensor management should be determined without having Z, it is necessary to find a way of hedging the objective function against this uncertainty. In [11, 1], it is suggested to mae a particular choice Z o for the unnowable Z, and Z o is a predicted ideal measurement-set that would be collected if there were no sensor noise. Thus PENT objective function is defined by Nx =Nx,Z o. 9 Based on 6, PENT objective function can be formulated as: Nx = υ 1 [1 p D, ] + n υ 1 [ p D, L zi ] p D, x i κ z i +υ 1 [ p D, L zi ], 1 where υ 1 [h] = h ξ υ 1 ξdξ, n is the predicted target number, x i } n are the predicted target states, p D, = 1 s l=1 1 p l D, x, x l is the s-sensor detection probability, L z = s l=1 g,l z l x, x l is the s-sensor lielihood function, and κ z i is the clutter intensity in s-sensor case, which can be modeled as s s l=1 rl l=1 cl zl i r and c are defined in Section II-B1. B. Proposed Cluster Scheduling Method 1 Particle PHD Filtering Implementation: In the proposed method, Particle PHD filtering implementation is applied to calculate the PENT value obtained by using each cluster. The prediction step of PHD recursion yields the } predicted particles and the associated weights x i, L 1 +J ωi 1. With them, υ [h] in Equation 1 can be presented as υ [h] = L 1 +J h x i ωi According to the single target transition density f 1 ζ and the spontaneous birth intensity γ defined in Equation 7, the target states at time are predicted as x j 1 f 1 x j 1, j =1,...,n, γ, j = n +1,...,n+ n γ,, } n 1 where x j 1 are the estimated target states at time 1, j=1 and n is the estimated target number at time 1. The number of each predicted target state can be predicted as 1, j =1,...,n; q j = p j n γ,, j = n +1,...,n+ n γ,, 13 where p i γ, and n γ, are defined in Equation 7. Then PENT value can be calculated based on Equation 1. This procedure is summarized in Table II. 196

4 Table II CALCULATION OF PENT VALUE USING PARTICLE PHD FILTER Table IV CALCULATION OF POSTERIOR INTENSITY ENTROPY For the c th cluster of s sensors } Given x i, L 1 +J } ωi, x j n 1 1, and the future j=1 joint sensor state x =x 1,...,x s. 1. According to Equations 1and 13, obtain the predicted target } states and the associated state numbers x j 1,q n+nγ, j. j=1. For each pair x j 1, x l, calculate the predicted ideal measurement zj l = η x j 1, x l,wherel =1,...,s. 3. Calculate PENT value obtained by the c th cluster: N c = L 1 +J 1 p D, x i ω i 1 + n+n γ, j=1 p D, x j G jx 1 κ z j +G jx q j, where G j x L 1 +J = p D, x i L zj ω i 1 L zj = s l=1 gl zj l xi, x l. Table III SELECTION OF ACTIVE CLUSTERS 1. Follow Table II to obtain the PENT value obtained by each cluster.. Choose the maximum one and calculate the quality vector d according to Equation Compare di with d TH and record the indices of quality measures of d that are less than the threshold. 4. The clusters with these indices are chosen for data fusion. Clusters Selection: In order to balance the estimation accuracy and the computational complexity, the number of active clusters are determined dynamically based on the quality of their estimation. The quality is measured by a vector d, which presents relatives between the PENT values Ni obtained by individual clusters and the maximum N max among all the PENT values: di = N max Ni 1 i C, 14 N max where C is the number of available clusters. When the value of di is small, the estimation quality of the cluster obtaining the maximum PENT value is equally worse as that of other clusters. Therefore, more clusters need to be included for data fusion. A threshold, d TH,issetfordi to determine whether or not to include more clusters. This method is described in Table III. IV. PROPOSED DATA FUSION METHODS Since multiple clusters are active at each time step, the procedure of data fusion is required. In this paper, a synchronized data fusion method and a sequential data fusion method are proposed for the first and second levels of hierarchy respectively. A. First-level Data Fusion Method Within each active clusters, sensors transmits their measurement-sets to the CH, which evaluates the quality of the measurement-sets and only includes the ones with higher For the l th sensor, 1. Follow the update step in Table I to obtain } the updated weights ω i L 1 +J l,,. Normalize the weights to get: } } ˆω i L 1 +J ω i L 1 +J l, l, = L 1 +J ω i 3. Calculate the posterior intensity entropy: Hl L 1+J ˆω i l, log ˆωi l,, information gain for data fusion. By doing this, the effect of the sensors with missed detection is efficiently reduced, and therefore the problem mentioned in [16] is fixed. The procedures of measurements evaluation and data fusion are described in the following subsections. 1 Measurement-set Evaluation: In the proposed architecture, the evaluation criterion is based on the information gain of the additional measurement-set Z [l] from the l th sensor, which can be expressed as follows: Φυ x H Z 1 H Z 1,Z [l], 15 where υ x is the posterior intensity, H denotes the posterior intensity entropy, and Z 1 denotes the measurement-set collected at time -1. From Equation 15, it can be seen that choosing the sensors with higher information gain Φυ x is equivalent to choose the ones with smaller posterior intensity entropy H Z 1,Z [l], which can be calculated based on the posterior intensity 6. This procedure is described in Table IV. Data Fusion: After the procedure of evaluation, multiple measurement-sets with higher information gain are selected to be used for data fusion in the first level of hierarchy. The proposed method is based on the fact that the predicted weights in Particle PHD filtering algorithm can be normalized and present for the probabilities of the corresponding particles. In this method, the associated estimate of target number is achieved by calculating the mean of the number estimates using individual measurement-sets. The procedure of first-level data fusion is described in Table V. This fusion method can efficiently increase the estimate accuracy, but there is another problem arises. After implementing step 3 in Table V, the particles corresponding to some true states may have trivial weights, and are eliminated during the resampling step. This will result in the missed detection of these true states. We can prevent this problem by implementing a modified resampling algorithm, in which no particles are eliminated unless the expected target number decreases. This proposed resampling algorithm is outlined in Table VI. B. Second-level Data Fusion In the second level of hierarchy, the estimate is updated sequentially at the CH of each active cluster, and the final updated result is transmitted to the BS. In sequential data 197

5 Table V FIRST-LEVEL DATA FUSION 8 6 Sensor CH BS 1. For each selected measurement-set Z [l], l =1,...,Q, follow the update step in Table I to obtain the update operator ω il =Ψ l w i 1,where [ Ψ l = 1 p D, x i + z Z [l] calculate the estimate of target number Ñ l = L 1 +J ω il, x i κ z+c z ψ,z. Obtain the associated estimate of target number Q l=1 N = round Ñ l. Q 3. Calculate the associated weight of each predicted particle: ω i = Q l=1 Ψ l ω i 1 }, L 1 ω i +J 4. Resample L 1 +J, x i ω i } L ω i to get L 1 +J,x i ω i. Table VI MODIFIED RESAMPLING ALGORITHM Given the predicted particles, the associated weights, the previous and current estimated target numbers N 1, N. 1. According to the standard resampling algorithm, obtain an array of indexes im} L 1+J m=1, which shows how many times each predicted particle is replicated.. If N N 1, replace the zero components of im} as 1s; 3. Update the particles based on the array of indexes, and the length of the updated particles has changed to be L 1 +J m=1 im. fusion, the order of update influences the final result, and the update with most reliable CH should be done first. The fusion order of CHs in the proposed architecture can be determined based on their PENT values, which are calculated in the cluster scheduling procedure. The CH with maximum PENT value should be used for update first. Furthermore, in order to reduce the degeneracy problem in Particle PHD filter, the updated particles are estimated Table VII PROPOSED DATA FUSION METHOD 1. Order the CHs based } on their PENT values, and obtain a sequence CH1,...,CH Q,whereCH1 has the maximum PENT value.. At the c th CH, where c =1,...,Q } Receive m l N 1 1,Pl 1. l=1 For l =1,...,N 1, draw ρ particles according to N ; m l 1,Pl 1, and assign equal weights ω i l = 1 ρ, Follow the Table V and obtain the locally updated } particles x i Lc, c, and the estimated target number ˆN c,, Partition the updated particles into ˆN c, clusters, and calculate the mean m l l c, and covariance P c, of each cluster. } Transmit m l ˆNc, c,,pl c, to the next CH or the BS. l=1 ], Y Coodinates m Figure 1. X Coordinates m The two-hierarchical architecture. with a Gaussian Mixture Model GMM [14], and only the parameters of GMM are transmitted. The procedure of the second-level data fusion is described in Table VII. V. SIMULATION RESULTS For illustration purpose, we consider a bearing and signal strength tracing in clutter environment. The two-tier hierarchical architecture is shown in Figure 1. There are eight available clusters in this simulation, and the BS is deployed at the origin. Each target moves according to the linear Gaussian dynamics in 16, x = 1 t 1 1 t 1 x 1 + Qu 1. where t 3 t 1/ 3 Q = t t q t 3 t, 3 t t and x = [ x,y, ẋ, ẏ ] T ; [x,y ] T denotes the target position at time, while [ẋ, ẏ ] T denotes the velocity at time, and t =1is the sampling period. The process noise u is a zero-mean Gaussian white noise with unit variance, and the factor q is used to control the intensity of the process noise. The initial target states are set to be X 1 = [5, 5,, ] T and X =[ 5, 5,, ] T. Existing Targets can survive with probability p S, =.98. For simplicity, no spawning birth is considered in this example, and new spontaneous birth of the targets is according to a Poisson point process with intensity function γ =.1N ; m 1 γ,p γ +.1N ; m γ,p γ, where m 1 γ = [15, 15,, ] T, m γ = [ 15, 15,, ] T, and P γ = diag [1, 1, 5, 5] T. For the sensor located at L =[l 1,l ] T, measurement model 198

6 Target Number True number of Targets Estimated Number of Targets X Coordinate m True Tracs True Tracs Y Coordinate m Figure. True and estimated target number p D,max =.99, κ = 1/91.34π Figure 4. x-y coordinates of estimated positions p D,max =.99, κ = 1/91.34π. Y Coordinate m True Trajectories Estimated Trajectories X Coordinate m Figure 3. True and estimated trajectories p D,max =.99, κ = 1/91.34π. can be formulated as follows: arctan x l1 z = y l S min S o + b, od o [x,y ] T L + b + ε, 16 where the second dimension of the measurement is formulated using an acoustic energy attenuation model [18, 19], with the signal strength S o = Energyunits, the reference distance d o =.3 meters, and the signal strength bias b =61.45 Energyunits, the measurement noise ε N ;,R with R = diag [ σθ T s],σ, σθ =.16π rad and σ s = Energyunits. The detection probability of each sensor is modeled as: p D, = p D,max.3 d/4, where p D,max is the optimal detection probability that can be achieved, and d is the Euclidean distance between one of the targets and this sensor. Clutter is uniformly distributed over the surveillance region [ π/, π/]rad [61.45, 35.79] Energyunit with an average rate of r points per scam, i.e. the intensity of clutter RFS κ = r/91.34π radenergyunit 1. In the followings, we present the simulation results given different clutter intensities κ and optimal detection probabilities p D,max. For the case that p D,max is.99, the threshold for choosing active clusters is set to be.3, and the threshold is set to be.5 for the case that p D,max is.95. Figure shows the true and estimated target number given the optimal observation probability p D,max =.99 and the clutter intensity κ =1/91.34π. The estimated positions and the true trajectories in this case are shown in Figure 3, and the individual x and y coordinates of the true tracs and estimated positions are shown in Figure 4. The above results are the performance for one trial. To evaluate the efficiency of the proposed architecture, the measurement of average performance are required. As supposed in [], the Wasserstein distance can be used as a multi-target miss-distance. The Wasserstein distance is defined for any two non-empty subsets ˆX, X as d p ˆX,X = min C ˆX X p C i,j ˆx i x j p, 17 j=1 where the minimum is taen over the set of all transportation matrices C, a transportation matrix is one whose entries C i,j satisfy C i,j, X j=1 C i,j =1/ ˆX, ˆX j=1 C i,j =1/ X. We use the mean Wasserstein miss-distance as the average performance criterion, and the simulation result over 5 trials in this case is shown in Figure 5. The simulation results given the optimal observation probability p D,max =.99 and the clutter intensity κ =5/91.34π are shown in Figure 6 - Figure 9. The results given p D,max =.95 and κ =1/91.34π are shown in Figure 1 - Figure 13. The simulation results in different cases shows that the estimation reliability can be guaranteed in an acceptable level, although denser clutter and poor quality of sensors cause the degradation in performance of the proposed architecture. This is quantified by the curves of mean Wasserstein miss-distance, which exhibit pea at the instances where the target number estimate is incorrect. As we can see in Figures 5, 9 and 13, the occurrence rate of incorrect number estimate is low. It is also clear that the mean miss-distance is acceptable when the estimated number is correct. For instance, in Figure 9, the 199

7 True Tracs True Tracs Mean Miss distance m X Coordinate m Y Coordinate m Figure 5. Average Tracing Performance p D,max =.99, κ = 1/91.34π. Figure 8. x-y coordinates of estimated positions p D,max =.99, κ = 5/91.34π. 4.5 True number of Targets Estimated Number of Targets Target Number Mean Miss distance m Figure 6. True and estimated target number p D,max =.99, κ = 5/91.34π. Figure 9. Average Tracing Performance p D,max =.99, κ = 5/91.34π True Trajectories Estimated Trajectories True number of Targets Estimated Number of Targets Y Coordinate m 1 Target Number X Coordinate m Figure 7. True and estimated trajectories p D,max =.99, κ = 5/91.34π. Figure 1. True and estimated target number p D,max =.95, κ = 1/91.34π. 191

8 True Tracs True Tracs Y Coordinate m True Trajectories Estimated Trajectories X Coordinate m Figure 11. True and estimated trajectories p D,max =.95, κ = 1/91.34π. mean miss-distance is under 65 m when the estimated number is correct. Since the second dimension of the measurement model in the simulation is severely non-linear, it is confirmed that the proposed architecture can address the MTT problem even when the measurement process is severely non-linear. VI. CONCLUSIONS In this paper, the MTT problem is addressed in a proposed two-tier hierarchical architecture, which considers a more realistic detection probability model. Efficient data fusion methods and scheduling scheme are implemented. The simulation results show that the proposed architecture wors well even when the measurement process is severely non-linear. Further research is being carried out in developing more efficient scheduling schemes and data fusion methods to improve the estimation accuracy and reduce energy consumption. X Coordinate m Y Coordinate m Figure 1. x-y coordinates of estimated positions p D,max =.95, κ = 1/91.34π. Mean Miss distance m Figure 13. Average Tracing Performance p D,max =.95, κ = 1/91.34π. REFERENCES [1] Y. Bar-Shalom and T.E. Fortmann, Tracing and Data Association. Boston, USA: Academic Press, [] Y. Bar-Shalom and X.R. Li, Multitarget-Multisensor Tracing, Principles and Techniques. Storrs, CT: YBS Publishing, [3] K. Panta and B. Vo, An Efficient Trac Management Scheme For The Gaussian-Mixture Probability Hypothesis Density Tracer, Int. Conf. Intelligent Sensing and Information Processing, Bangalore India, 6. [4] K. Panta, B. Vo, and S. Singh, Novel Data Association Schemes for the Probability Hypothesis Density Filter, IEEE Transaction on Aerospace and Electronic Systems, 7. [5] R. Mahler, An Introduction to Multisource-Multitarget Statistics and Applications. Locheed Martin Technical Monograph,. [6] R. Mahler, Multi-target Bayes filtering via first-order multi-target moments, IEEE Trans. AES, vol. 39, no. 4, pp , 3. [7] B. Vo, S. Singh, and A. Doucet, Sequential Monte Carlo Methods for Multi-target Filtering with Random Finite Sets, IEEE Trans. AES, vol. 41, no. 4, pp , 5. [8] B. Vo and W. Ma, The Gaussian Mixture Probability Hypothesis Density Filter, IEEE Trans. Signal Processing, vol. 54, no. 11, pp , 6. [9] R. Mahler, Multitarget Filtering via First-order Multitarget Moments, IEEE Trans. Aerospace and Electronics Sys., vol. 39, no. 4, pp , 3. [1] R. Mahler, Sensor Management with Non-ideal Sensor Dynamics, Proc. 7 th Int l Conf. on Information Fusion, 4. [11] R. Mahler, Target Preference in Multitarget Sensor Management: A Unified Approach, Proc. SPIE, vol. 549, 4. [1] R. Mahler, Unified Sensor Management Using CPHD Filters, Proc. 1 th Int l Conf. on Information Fusion, pp. 1-7, 7. [13] B. Vo, S. Singh, and W. K. Ma, Tracing Multiple Speaers with Random Sets, ICASSP, Montreal, Canada, 4. [14] S. Xiaohong and H. Yu-Hen, Distributed Particle Filters for Wireless Sensor Networ Target Tracing, ICASSP, vol. 4, 5. [15] W. K. Ma, B. Vo, S. Singh, and A. Baddeley, Tracing an Unnown Time-varying Number of Speaers Using TDOA Measurements, A Random Finite Set Approach, IEEE Trans. Signal Processing, vol. 54, no. 9, pp , 6. [16] O. Erdinc, P. Willett, and Y. Bar-Shalom, Probability Hypothesis Density Filter for Multitarget Multisensor Tracing, Proc. 8 th Int l Conf. on Information Fusion, vol. 1, 5. [17] T. ShuRong, H. You, and Y. XiaoShu, Distributed Multi-sensor Multitarget Tracing with Random Sets I, ICNC, vol. 3, pp , 7. [18] A. Swain, Characterization of Acoustic Sensor Motes for Target Tracing in Wireless Sensor Networs. Masters thesis, Arizona State University, 6. [19] M. Vemula, M.F. Bugallo, and P.M. Djuric, Target Tracing in a Two- Tiered Hierarchical Sensor Networ. Acoustics, ICASSP, 6. [] J. Hoffman and R. Mahler, Multitarget Miss Distance via Optimal Assignment, IEEE Trans. Sys., Man, and Cybernetics-Part A, vol. 34, no. 3, pp ,