Multi-Target Tracking in a Two-Tier Hierarchical Architecture

Multi-Target Tracking in a Two-Tier Hierarchical Architecture Jin Wei

Xudong Wang

Vassilis L. Syrmos

Department of Electrical Engineering University of Hawaii at Manoa Honolulu, HI 96822, U.S.A. Email: [email protected]

Department of Electrical Engineering University of Hawaii at Manoa Honolulu, HI 96822, U.S.A. Email: [email protected]

Department of Electrical Engineering University of Hawaii at Manoa Honolulu, HI 96822, U.S.A. Email: [email protected]

Abstract—In this paper, a two-tier hierarchical architecture is proposed to address the multi-target tracking 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 works 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 tracking.

Keywords: Particle Probability Hypothesis Density filter, Data Fusion, cluster scheduling, Gaussian Mixture Model. I. I NTRODUCTION Multi-Target Tracking (MTT) has received many attentions these years. It involves joint estimation of unknown and time-varying number of targets and their states in cluttered environment [1, 2, 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 framework 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 drawback 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 works 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 tracked 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 2005, Mahler derived an objective function, called the Posterior Expected Number of Targets (PENT), for multi-target sensor management based on the PHD filter [9, 10, 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 [12]. 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 know, 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

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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 tracking are presented in section V. Finally, some conclusions and potential extensions are discussed in section VI. II. P ROBLEM F ORMULATION A. Random Finite Set Model for Multi-Target Tracking The MTT problem can be modeled by RFS framework [5]. Let χ be the single target state space, and M (k) be the number of targets at time k, then multi-target state at time k is presented by Xk = {xk,1 , xk,2 , . . . , xk,M (k) } ∈
(χ), where
(χ) denotes the collection of all finite subsets of χ. For a multi-target state Xk−1 at time step k − 1, each xk−1 ∈ Xk−1 either continues to exist at time k with probability pS,k (xk−1 ), or dies with probability 1 − pS,k (xk−1 ). Therefore, for a given state xk−1 ∈ Xk−1 at time k − 1, its behavior at the next time step can be modeled as the RFS Sk|k−1 (xk−1 ), that can take on either {xk } when the target survives, or φ when it dies. A new target at time k can arise either by spontaneous births which can be modeled by Γk , or by spawning from xk−1 which can be modeled by Bk|k−1 (xk−1 ). Given a multi-target state Xk−1 at time k − 1, the multi-target state Xk at time k is formed by union of the surviving targets and new targets,         Sk|k−1 (ζ)  Bk|k−1 (ζ) Γk , Xk =  ζ∈Xk−1

ζ∈Xk−1

(1) Similarly, let the measurement-set collected by the i sen[i] sor at time k to be Zk ∈
(Z[i] ). A given target state xk ∈ Xk is either detected with probability pD,k (xk ) or missed with probability 1−pD,k (xk ). Consequently, the measurement from [i] xk at sensor i can be modeled by the RFS Θk (xk ), that can take on either {zk } when the target is detected, or φ otherwise. [i] The ith sensor can also receive a set Ck of clutter. So, given a multi-target state Xk at time k, the measurement-set collected by the ith sensor is formulated by    [i]  [i] [i] Zk = Θk (x) Ck , (2) th

x∈Xk

Let Q be the number of sensors, then the RFS of measurements at time k is modeled by  [1] [2] [Q] . (3) Zk = Zk , Zk , . . . , Zk The RFS Xk encapsulates all aspects of multi-target tracking problem, and Zk encapsulates all sensor characteristics. The multi-target tracking can be posed as follows: given the measurement-set Z1:k collected from sensors up to time k, the ˆ k that is the expectation of the posterior problem is to find X density function p(Xk |Z1:k ).

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 υ(x) is defined as follows. For a RFS X on χ with probability distribution P , the integral of υ(x) over region S gives the expected number of elements of X that are in S,

 

  υ(x)dx , (4) N = X S  P (dX) = S

It can be shown that the posterior intensity can be propagated in time via the PHD recursion.

υk|k−1 (x)

= +



pS,k (ζ)fk|k−1 (x|ζ)υk−1 (ζ)dζ βk|k−1 (x|ζ)υk−1 (ζ)dζ + γk (x), (5)

υk (x) = [1 − pD,k (x)]υk|k−1 (x)

pD,k (x)gk (z|x)υk|k−1 (x)  + κk (z) + pD,k (ξ)gk (z|ξ)υk|k−1 (ξ)dξ z∈Z k

(6) where fk|k−1 (·|ζ) denotes the single target transition density, γ(·) denotes the intensity of the spontaneous birth RFS, βk|k−1 (·|ζ) denotes the intensity of the spawning birth RFS, pS,k (ζ) denotes the probability that a target continues to exist given that its previous state is ζ, gk (·|x) denotes the single target measurement likelihood, pD,k (x) denotes the detection probability given a state x, and κk (·) denotes the intensity of the clutter RFS. Note that κk (·) can be modeled as rk ck (·), where rk is the average number of clutter points per scan and ck 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 peaks of PHD. 2) 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 Γk can be modeled as a Gaussian mixture of the form nγ,k 

(i)  (i) (i) (7) pγ,k N x; mγ,k , Pγ,k . γk (x) =

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i=1

Table I PARTICLE PHD F ILTERING A LGORITHM

collected if there were no sensor noise. Thus PENT objective function is defined by N (x∗k ) = N (x∗k , Zko ) .

1. Initialization
Draw L0 particles from the prior PHD and assign weights: (i) ω0 = LN , where N is the prior estimation of target number. 0 2. At time k ≥ 1   (i)
According to the proposal densities qk ·|xk−1 , Zk and pk (·|Zk) draw predicted particles: 

Based on (6), PENT objective function can be formulated as: N (x∗k )

(i)

qk ·|xk−1 , Zk , i = 1, . . . , Lk−1 ; pk (·|Zk ) , i = Lk−1 + 1, . . . , Lk−1 + Jk . and calculate the associated weights as:  (i) (i) φk|k−1 (˜ xk ,xk−1 ) (i)   ωk−1 , i = 1, . . . , Lk−1 ,  (i) (i) qk (˜ xk |Zk ) w ˜k|k−1 = (i)  γk (˜ xk )   J1 , i = Lk−1 + 1, . . . , Lk−1 + Jk , (i) (i)

x ˜k ∼

k

=

+
Update each of the weights 

  (i) (i) ˜k |xk−1 βk|k−1 x (i)

(i)

 where υk|k−1 [h] = h (ξ) υk|k−1 (ξ)dξ, n is the predicted n predicted target states, p˘D,k = target number, {xi }i=1 are the   s  l ∗l is the s-sensor detection prob1 − l=1 1 − pD,k x, xk  l  s is the s-sensor likelihood ability, Lz = l=1 gk,l z |x, x∗l k function, and κk (zi ) is theclutter intensity case,  s inl s-sensor  s l l which can be modeled as l=1 rk l=1 ck (zi ) (rk and ck are defined in Section II-B1).

(i)

pS,k (xk−1 )fk|k−1 (˜ xk |xk−1 ).

   (i) ˜ ψk,z x  (i) (i) (i) ˜k|k−1 , xk ) + z∈Zk κ (z)+Ck (z) w ω ˜ k = 1 − pD,k (˜ k k   (i) (i) (i) ˜k = pD,k (˜ xk )gk (z|˜ xk ) where ψk,z x    Lk−1 +Jk (j) (j) ˜k w ˜k|k−1 . ψk,z x Ck (z) = j=1  (j) ˜k|k = Lk−1 +Jk ω ˜ ,
Calculate the total mass N  (i)  Lk−1 +Jk j=1  (i)k  Lk ω ˜k ωk (i) (i) resample ,x ˜k to get , xk , ˜ ˜ N N k|k k|k i=1 i=1  Lk  ˜k|k to get ω (i) , x(i) . rescale the weights by N k k

B. Proposed Cluster Scheduling Method

i=1

(i)

(i)

= υk|k−1 [1 − p˘D,k ] n

pD,k Lzi ] υk|k−1 [˘ , + p˘D,k (xi ) · κ (z ) + υ pD,k Lzi ] k i k|k−1 [˘ i=1 (10)

pk (˜ xk |Zk )

where (i) (i) φk|k−1 (˜ xk , xk−1 )

(9)

(i)

where nγ,k , pγ,k , mγ,k , Pγ,k , i = 1, . . . , nγ,k , are given model parameters that determine the shape of the intensity.

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 thepredicted (i)

III. P ROPOSED C LUSTER S CHEDULING M ETHOD

Lk−1 +Jk

υ [h] =

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, 10, 11]. The basic idea of this objective function is outlined as follows. ∗s Let x∗k = (x∗1 k , . . . , xk ) be the future joint sensor state, then an objective function O(x∗k , Zk ) for sensor management can be constructed from the data-updated intensity υk (x|Zk ). The objective function mathematically describes what goals we want sensor management to maximize. Equation (4) provides the following choice for objective function:

υk (x, x∗k |Zk )dx . (8) N (x∗k , Zk ) =

Lk−1 +Jk

(i)

(i)

h(˜ xk )˜ ωk|k−1 .

.

(11)

i=1

According to the single target transition density fk|k−1 (·|ζ) and the spontaneous birth intensity γ (·) defined in Equation (7), the target states at time k are predicted as  xjk|k−1

∼

fk|k−1 (·|xjk−1 ), γ(·),

n  where xjk−1

j=1

j = 1, . . . , n, j = n + 1, . . . , n + nγ,k , (12) are the estimated target states at time k −1,

and n is the estimated target number at time k−1. The number of each predicted target state can be predicted as 

S

Since the sensor management should be determined without having Zk , it is necessary to find a way of hedging the objective function against this uncertainty. In [11, 12], it is suggested to make a particular choice Zko for the unknowable Zk , and Zko is a predicted ideal measurement-set that would be

(i)

˜ k|k−1 particles and the associated weights x ˜k , ω i=1 With them, υ [h] in Equation (10) can be presented as

qj = (i)

1, j = 1, . . . , n; (j−n) pγ,k , j = n + 1, . . . , n + nγ,k ,

(13)

where pγ,k and nγ,k are defined in Equation (7). Then PENT value can be calculated based on Equation (10). This procedure is summarized in Table II.

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Table II C ALCULATION OF PENT VALUE U SING PARTICLE PHD F ILTER For the cth cluster of s sensors   Lk−1 +Jk  n (i) (i) Given x ˜k , ω ˜ k|k−1 , xjk−1 i=1

j=1

Table IV C ALCULATION OF POSTERIOR INTENSITY ENTROPY For the lth sensor, 1. Follow the update step in Table I to obtain   (i) Lk−1 +Jk , the updated weights ωl,k i=1 2. Normalize the weights to get:  Lk−1 +Jk  (i)   ωl,k (i) Lk−1 +Jk =  Lk−1 +Jk (i) ω ˆ l,k

, and the future

∗s joint sensor state x∗k = (x∗1 k , . . . , xk ). 1. According to Equations (12)and (13), obtain the predicted target  n+nγ,k  . states and the associated state numbers xjk|k−1 , qj j=1   j ∗l 2. For each pair xk|k−1 , xk , calculate the predicted ideal   measurement zjl = ηk xjk|k−1 , x∗l k , where l = 1, . . . , s.

i=1

i=1

ωk

3. Calculate the posterior intensity entropy:  Lk−1 +Jk (i) (i) ω ˆ l,k log ω ˆ l,k , H(l)
− i=1

3. Calculate PENT value obtained by the cth  cluster:  Lk−1 +Jk  (i) (i) 1 − p˘D,k x ˜k ω ˜ k|k−1 N (c) = i=1   ∗  n+nγ,k G (x j k) qj , + p˘D,k xjk|k−1 j=1 κk(zj )+Gj (x∗ k)     Lk−1 +Jk (i) (i) ˜k Lzj ω where Gj x∗k = i=1 p˘D,k x ˜ k|k−1    Lzj = sl=1 gkl zjl |˜ x(i) , x∗l k .

i=1

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 [l] of the additional measurement-set Zk from the lth sensor, which can be expressed as follows:   [l] (15) Φ (υk (x))
H (Zk−1 ) − H Zk−1 , Zk ,

Table III S ELECTION OF ACTIVE C LUSTERS

1. Follow Table II to obtain the PENT value obtained by each cluster. 2. Choose the maximum one and calculate the quality vector d according to Equation (14). 3. Compare d(i) with dTH 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.

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.

where υk (x) is the posterior intensity, H(·) denotes the posterior intensity entropy, and Zk−1 denotes the measurement-set collected at time k-1. From Equation (15), it can be seen that choosing the sensors with higher information gain Φ (υk (x)) is equivalent to choose the ones with smaller posterior intensity entropy [l] H Zk−1 , Zk , which can be calculated based on the posterior intensity (6). This procedure is described in Table IV. 2) 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.

A. First-level Data Fusion Method

B. Second-level Data Fusion

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

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

2) 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 N (i) obtained by individual clusters and the maximum Nmax among all the PENT values: d(i) =

Nmax − N (i) Nmax

1 ≤ i ≤ C,

(14)

where C is the number of available clusters. When the value of d(i) 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, dTH , is set for d(i) to determine whether or not to include more clusters. This method is described in Table III. IV. P ROPOSED DATA F USION M ETHODS

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Table V F IRST- LEVEL DATA F USION

800 Sensor CH BS

600

[l]

1. For each selected measurement-set Zk , l = 1, . . . , Q,
follow the update step in Table I to obtain (i)

= Ψl · w ˜k|k−1 , where 

(i)

Ψl = 1 − pD,k (˜ xk ) +

[l]

z∈Zk

  (i) ˜k ψk,z x

κk (z)+Ck (z)

Y Coodinates (m)

(il )

the update operator ω ˜k 

400

 ,


calculate the estimate of target number (il ) ˜ (l) =  Lk−1 +Jk ω N ˜k , i=1 k 2. Obtain the associated estimate of target number  Nk = round

−600

−800 −800

.

Q

(i)

i=1

 to get

ω ˜k  Lk−1 +Jk (i)

ωk

 Lk−1 +Jk i=1

−400

−200

0

Figure 1.

,x ˜k

 Lk

i=1

i=1

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

800

The two-hierarchical architecture.

For illustration purpose, we consider a bearing and signal strength tracking 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),   1 ∆t 0 0  0 1 0 0   xk =   0 0 1 ∆t  xk−1 + Quk−1 . 0 0 0 1 where Q=

  √  q 

∆t3 3 ∆t2 2

0 0

∆t2 2

∆t 0 0

0 0

∆t3 3 ∆t2 2

T

Table VII P ROPOSED DATA F USION M ETHOD

1. !Order the CHs based " on their PENT values, and obtain a sequence CH1 , . . . , CHQ , where CH1 has the maximum PENT value. 2. At the cth CH, where c = 1, . . . , Q  Nk−1  (l) (l) . Receive mk−1 , Pk−1 l=1 , draw ρ particles according to
Forl = 1, . . . , Nk−1  (l) (l) (i) N ·; mk−1 , Pk−1 , and assign equal weights ωl = ρ1 ,
Follow the Table V and obtain the locally updated   (i) Lc,k ˆc,k , particles x and the estimated target number N ˆc,k clusters, and calculate
Partition the updated particles into N (l) (l) the mean mc,k and covariance Pc,k of each cluster. ˆ  (l) (l) Nc,k to the next CH or the BS. Transmit mc,k , Pc,k l=1

600

V. S IMULATION R ESULTS

Given the predicted particles, the associated weights, the previous and current estimated target numbers Nk−1 , Nk . 1. According to the standard resampling algorithm, obtain an array Lk−1 +Jk , which shows how many times each of indexes {i(m)}m=1 predicted particle is replicated. 2. If Nk ≥ Nk−1 , replace the zero components of {i(m)} as 1s; 3. Update the particles based on the array of indexes, and the length  Lk−1 +Jk i(m). of the updated particles has changed to be m=1

i=1

400

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.

.

Table VI M ODIFIED R ESAMPLING A LGORITHM

c,k

200

(i)

(i)

ω ˜k

(i) , xk (i)

ωk

−600

X Coordinates (m)

3. Calculate the associated weight  of each predicted  (i) (i) Q ω ˜ k|k−1 , Ψ particle: ω ˜k = l l=1  Lk−1 +Jk  4. Resample

0

−200

−400

Q

˜ (l) N k l=1

200

0 0

∆t2 2

1/2    

,

∆t

T

and xk = [ xk , yk , x˙ k , y˙ k ] ; [xk , yk ] denotes the target T position at time k, while [x˙ k , y˙ k ] denotes the velocity at time k, and ∆t = 1 is the sampling period. The process noise uk 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 X01 = [250, 250, 0, 0]T and X02 = [−250, −250, 0, 0]T . Existing Targets can survive with probability pS,k = 0.98. 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  (1) (2) function γk = 0.1N ·; mγ , Pγ + 0.1N ·; mγ , Pγ , where (1)

(2)

T T mγ = [150,  150, 0, 0] , mTγ  = [−150, −150, 0, 0] , and Pγ = diag [100, 100, 25, 25] . For the sensor located at L = [l1 , l2 ]T , measurement model

1908

800

3.5

Target Number

True Tracks Position Estimates

600

True number of Targets Estimated Number of Targets

X Coordinate (m)

400

200

0

−200

−400

3

−600

−800

−1000

0

5

10

15

20

25

30

35

40

25

30

35

40

Time Step

600 True Tracks Position Estimates

500

2.5

Y Coordinate (m)

400

2

300 200 100 0 −100 −200

0

5

10

15

20

25

30

35

40

−300

Time Step

−400

0

5

10

15

20

Time Step

Figure 2. True and estimated target number (pD,max = 0.99, κ = 10/291.34π).

Figure 4. x-y coordinates of estimated positions (pD,max = 0.99, κ = 10/291.34π).

600 True Trajectories 500

Estimated Trajectories

400

Y Coordinate (m)

300

200

100

0

−100

−200

−300

−400 −1000

−800

−600

−400

−200

0

200

400

600

X Coordinate (m)

Figure 3. True and estimated trajectories (pD,max = 0.99, κ = 10/291.34π).

can be formulated as follows:     −l1 arctan xykk −l2   + εk ,  zk =  So do + b min So + b, [xk ,y T k ] −L

is set to be 0.5 for the case that pD,max is 0.95. Figure 2 shows the true and estimated target number given the optimal observation probability pD,max = 0.99 and the clutter intensity κ = 10/291.34π. 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 tracks 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 [20], the Wasserstein distance can be used as a multi-target miss-distance. The Wasserstein distance is defined for any two non-empty subsets ˆ X as X, + , ˆ |X| |X| ,  

, p ˆ Ci,j ˆ xi − xj p , dp X, X = min C

(16)

where the second dimension of the measurement is formulated using an acoustic energy attenuation model [18, 19], with the signal strength So = 291.34 Energyunits, the reference distance do = 0.3 meters, and the signal strength bias b = 61.45 Energyunits, the measurement noise ε ∼ N (·; 0, Rk ) with *T ) Rk = diag( σθ2 , σs2 ), σθ = 0.016π rad and σs = 2 Energyunits. The detection probability of each sensor is modeled as: pD,k = pD,max − 0.03 × d/400, where pD,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/2, π/2]rad × [61.45, 352.79] Energyunit with an average rate of r points per scam, i.e. the intensity of clutter −1 RFS κ = r/291.34π (radEnergyunit) . In the followings, we present the simulation results given different clutter intensities κ and optimal detection probabilities pD,max . For the case that pD,max is 0.99, the threshold for choosing active clusters is set to be 0.3, and the threshold

(17)

i=1 j=1

where the minimum is taken over the set of all transportation matrices C, (a transportation matrix is one whose entries Ci,j |X| ˆ ˆ |X| satisfy Ci,j ≥ 0, j=1 Ci,j = 1/|X|, j=1 Ci,j = 1/|X|). We use the mean Wasserstein miss-distance as the average performance criterion, and the simulation result over 50 trials in this case is shown in Figure 5. The simulation results given the optimal observation probability pD,max = 0.99 and the clutter intensity κ = 50/291.34π are shown in Figure 6 - Figure 9. The results given pD,max = 0.95 and κ = 10/291.34π are shown in Figure 10 - 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 peak 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

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Figure 5. Average Tracking Performance (pD,max = 0.99, κ = 10/291.34π).

Figure 8. x-y coordinates of estimated positions (pD,max = 0.99, κ = 50/291.34π).

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Figure 9. Average Tracking Performance (pD,max = 0.99, κ = 50/291.34π).

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Figure 10. True and estimated target number (pD,max = 0.95, κ = 10/291.34π).

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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.

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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 works 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. R EFERENCES

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Figure 12. x-y coordinates of estimated positions (pD,max = 0.95, κ = 10/291.34π).

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Figure 13. Average Tracking Performance (pD,max = 0.95, κ = 10/291.34π).

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