A Novel MS-MeMBer Filter for Extended Targets Tracking

Conventional multi-sensor multi-target multi-Bernoulli (MS-MeMBer) filters are based on the assumption that each target produces at most one measurement per time step. However, this assumption is not always reasonable in practice as an extended target can generate multiple measurements per step due to the recent improvement in the sensor resolution. In this case, a potential estimation bias may occur in the current MS-MeMBer filters. Therefore, a novel extended target MS-MeMBer filter and its Gaussian inverse Wishart mixture implementation are given in this paper. Specifically, we modify the update process of the MS-MeMBer filter by assuming that the generation of extended target measurements follows an approximate Poisson-Body model. Simulation results validate that the proposed filter can effectively estimate the shape and position of the extended target.


I. INTRODUCTION
The random finite set (RFS) [1] has received much attention in the multi-target filtering domain due to its superiority in avoiding complicated data association steps [2]- [12]. Under the RFS framework, the target state estimation is transformed into a set-valued estimation problem. For singlesensor scenarios, the probability hypothesis density (PHD), the cardinalized PHD (CPHD), and the multi-Bernoulli filter were proposed since the optimal multiple target tracking is intractable [2]- [5]. The significant difference between the CPHD/PHD filter and the multi-Bernoulli filter is that the PHD and CPHD filters are moment-based approximations to the multi-target Bayes filter, while the multi-Bernoulli filter approximates the multi-target posterior as a multi-Bernoulli distribution. To obtain the target trajectories, B. T. Vo and B. N. Vo introduced a generalized labeled multi-Bernoulli (GLMB) filter, which can generate target track labels [6]. Subsequently, compared to [6], a more efficient labeled multi-Bernoulli (LMB) filter was designed in [7] by only allowing the propagation of a single set of track labels.
The associate editor coordinating the review of this manuscript and approving it for publication was Chengpeng Hao .
For two-sensor scenarios, Mahler [8] proposed an accurate PHD filter based on binary segmentations. To process measurements from an arbitrary number of sensors, a parallel-combination approximate multi-sensor (PCAM) CPHD/PHD filter was presented in [9] and then an exact multi-sensor CPHD/PHD (MS-CPHD/PHD) filter was given in [10]. In [11], Saucan introduced a multi-sensor multitarget multi-Bernoulli (MS-MeMBer) filter via a Gaussian mixture (GM) implementation. Similarly, a generalization of the GLMB filter for the multi-sensor case, named multi-sensor GLMB (MS-GLMB) filter, was proposed in [12].
The filters above obey the ''standard'' observation model, that is, one target generates at most one measurement per time step, and each measurement originates from at most one target. However, in the high-resolution sensor system, one target may occupy multiple resolution cells and thus can produce multiple measurements. Such targets are known as extended targets [13], [14]. For the extended target tracking, Gilholm et al. proposed an approximate Poisson-Body (APB) model [15], in which the measurement is assumed to follow a multi-dimensional Poisson process. Based on the APB model, an extended target PHD (ET-PHD) filter was proposed in [16], and a GM implementation of it was given in [17]. Extensions of the ET-PHD filter for handling the shape estimation by using the random matrix model (RMM) [18]- [21] were given in [22], [23], and the resulting filters are the Gaussian inverse Wishart PHD (GIW-PHD) filter and the gamma Gaussian inverse Wishart PHD (GGIW-PHD) filter. In [22], the target kinematical state was defined by a Gaussian distribution, while the target extension was assumed to follow an inverse Wishart distribution. As an improvement on the GIW-PHD filter, the GGIW-PHD filter can also obtain the estimation of target measurement rates. Besides, other filters based on the RFS, which include the gamma Gaussian inverse Wishart CPHD (GGIW-CPHD) filter, the extended target MeMBer filter, and the GGIW-GLMB/LMB filter, have also been introduced for the extended target case [24]- [27]. Extended models such as the random hypersurface model [28] and the star-convex [29] have been proposed as well for targets with more complex shapes.
The MS-CPHD filter and the MS-MeMBer filter are both two important methods for multi-sensor multi-target tracking, while the MS-MeMBer filter has lower computational requirements. For the extended target case, however, a serious potential estimation bias could occur in the MS-MeMBer filter. Therefore, in this paper, we present an extended target MS-MeMBer (ET-MS-MeMBer) filter as well as its GIW mixture implementation. Particularly, we first assume that the measurement of extended targets obeys an APB model, and then modify the update process of the original MS-MeMBer filter. As the proposed ET-MS-MeMBer filter requires the knowledge of all the partitions of multi-sensor measurements, here we introduce a greedy approach based on the two-step partition method [10], [11] and the distance partition method [17]. Simulation results show that the proposed ET-MS-MeMBer filter can effectively estimate the position and shape of the extended target.
An idea of iterative updating was introduced in [30], in which the updated result of the previous sensor is taken as an input to the update step of the next sensor. This realization makes it possible to generalize the single-sensor filters to multi-sensor scenarios. As the single-sensor GGIW-CPHD filter is an important method for extended target tracking, in this paper, we extend it to the iterated-corrector GGIW-CPHD (IC-GGIW-CPHD) filter for the multi-sensor case. Meanwhile, a comparison of the IC-GGIW-CPHD filter and the ET-MS-MeMBer filter is presented in this paper. Results show that with the number of sensors increases, the two filters obtain higher filtering accuracy despite a growing computational cost. Further, we demonstrate that the proposed ET-MS-MeMBer filter outperforms the IC-GGIW-CPHD filter.
The rest of this paper is organized as follows. In Section II, we present an overview of the APB model, the multi-Bernoulli RFS, and the MS-MeMBer filter. Our proposed ET-MS-MeMBer filter is derived in Section III. The GIW mixture implementation is provided in Section IV. Simulation experiments are presented in Section V. Conclusions are given in Section VI.

A. THE APB MODEL FOR EXTENDED TARGETS
In the scenario of tracking extended targets, a target may generate multiple measurements. Assume at time k, the measurement z generated by the target with state x on sensor i obeys the spatial distribution f x (z) = h i,k (z|x), and the number of extended measurements is Poisson-distributed with mean γ i (x). Then the likelihood function of x can be denoted by where Z is a collection of all measurements generated by target x on sensor i at time k, and |Z| is the cardinality of set Z.
The probability generation functional (PGFL) of f i,k (Z|x) is where p g (x) = g(z)h i,k (z|x)dz.

B. THE MULTI-BERNOULLI RFS
The Bernoulli RFS X B can be represented by the probability of existence r and the density p(x), then the distribution function is which can be further simplified to π (X B ) = (r, p). The PGFL of π (X B ) is The multi-Bernoulli RFS X mB is a union of a fixed number of independent Bernoulli RFSs. Given an RFS set X mB = {x 1 , . . . , x n }, the distribution function is where M is the number of Bernoulli components. In this paper, the abbreviation π(X mB ) = (r (j) , p (j) ) M j=1 is adopted for the multi-Bernoulli distribution function. The PGFL of π (X mB ) is VOLUME 8, 2020 The PHD of π (X mB ) is

C. THE MS-MEMBER FILTER
The prediction step of the MS-MeMBer filter is the same as that of the multi-Bernoulli filters [4], [5]. Suppose that at time k − 1, the posterior multi-target density is a multi-Bernoulli with and we make the following assumptions: • The target can survive to the next step with a probability of ρ sv ; • At time k, the target is born with a probability of r b,k and a spatial density of p b,k . Then, the predicted density is also a multi-Bernoulli with where M b,k is the number of newborn targets. The existence probability and the density of the survival target are given as follows where f k|k−1 (x|·) denotes the transfer function. Assume that the predicted multi-target density is a multi-Bernoulli with Then the updated density can be approximated by a multi-Bernoulli witĥ where each of the updated Bernoulli components r is given as follows P and U are partitioning hypotheses [11] of the multisensor measurements, Q is a union of all the hypotheses, and W j 1:s is the measurement set generated by the j-th Bernoulli component (r is the probability of the target with state x not being detected by any sensor, f (W j 1:s |x) is the multi-sensor likelihood function, and K P is a coefficient associated with P.

III. THE EXTENDED TARGET MS-MEMBER FILTER
In this section, we present the proposed ET-MS-MeMBer filter. Since no measurement information is used in the prediction process, the prediction step of the ET-MS-MeMBer filter is the same as that of the MS-MeMBer filter. Therefore, only the update step of the ET-MS-MeMBer filter is given in this section.

A. MULTI-SENSOR MEASUREMENT PARTITIONING
In the update process of the ET-MS-MeMBer filter, all the partitioning forms of the multi-sensor measurements need to be obtained. Measurement partitioning aims to divide a measurement set into a finite number of mutually disjoint subsets. The principle is to put homologous measurements into the same subset. Suppose that at time k, the measurement set generated by all sensors is Z 1:s = (Z 1 , · · · , Z s ), where each Z i contains all measurements from the i-th sensor. According to the number of the predicted Bernoulli components M k|k−1 , the set Z 1:s can be repartitioned as P = , where W 0 1:s is the clutter set from all sensors.
As the assumption that ''one target produces at most one measurement'' is invalid in the extended target case, we make the following definitions: where ⊕ is the disjoint union, and w 0 i contains all clutter measurements from the i-th sensor.
i } is the set of all measurements generated by target j on the i-th sensor.
. The element w i is allowed to be empty.
Based on the definitions above, we can group different measurements from the same sensor together. Define the mapping function as where w j i → i shows that the subset w j i is generated by target j from the i-th sensor.

B. ET-MS-MEMBER UPDATE
Based on the APB model, the updating equations for extended targets are introduced as follows. Detailed derivations are given in the Appendix.
Proposition 1: In the extended target scenario, the undetected probability of the target with state x is where p i,d (x) is the detection probability of the target by sensor i, and γ i (x) is the mean of the extended measurements. Proposition 2: At time k, the multi-sensor likelihood for the extended target is where Pr(n|λ) is the Poisson probability distribution with mean λ. c i,k (z) is the spatial distribution of clutter at sensor i, and Proposition 3: In the extended target scenario, the coefficient K P can be expressed as where C The other formulas for the ET-MS-MeMBer filter are the same as those for the MS-MeMBer filter.

IV. THE GIW MIXTURE IMPLEMENTATION
A closed-form solution to the ET-PHD filter has been established for GIW models in [22], where the kinematic state of the target is represented by a Gaussian distribution, and the target extension is described by an inverse Wishart distribution. In subsection IV-A, we present the basic form of the GIW model. The numerical implementation of the ET-MS-MeMBer filter is given in subsection IV-B and IV-C.

A. THE BASIC MODEL
Assume that at time k − 1, the target state x k−1 is composed of the kinematic statex k−1 and the target extension . Then the motion model of the target can be expressed as: where ⊗ is the Kronecker product, and I d is a d-dimensional identity matrix. N (·; m, P) is a Gaussian distribution with mean m and variance P. T s is the sampling period. σ is the scalar acceleration standard deviation and θ is the maneuver correlation time.
Assume that at time k, the target generates n k measurements, i.e., Z k = z . Then the observation model can be given by: where H k = [100], and e (j) k is the white Gaussian noise with covariance determined by the target extension.

B. THE NUMERICAL IMPLEMENTATION
Assume that each Bernoulli density in (9) has a GIW mixture form: where J  Given the probability density of newborn targets: b,n,k ). (32) Likewise, the predicted Bernoulli density has a GIW mixture form, and the legacy components become: where In the predicted degrees of freedom (37), τ is a temporal decay constant.
Assume that each predicted Bernoulli density in (13) has a GIW mixture form: n,k|k−1 ). (39) Set the detection probability and the extended measurement rate as constants, i.e., p i,d (x) = p i,d , γ i (x) = γ i . Then the updated Bernoulli density is calculated by where w (P,j) The right part of (43) is a recursive update operation, which follows After each update step, the mean m , covariance P , degrees of freedom v , and inverse scale matrix V are respectively The likelihood ratio q z (w j i ) in (44) is given by and d (·) is a multivariate Gamma distribution.

C. A GREEDY PARTITIONING MECHANISM FOR EXTENDED TARGETS
The exact solution for the ET-MS-MeMBer filter is computationally infeasible since all partitioning hypotheses are needed in the update step. Hence, in this section, we propose an efficient approximate solution based on the greedy mechanism. Firstly, all measurements from the same source are formed as a subset. For each measurement set Z i , define w as a subset of Z i , and w contains extended measurements generated by the same target. Define W (Z i ) as the collection of all subsets. For example, if the measurement set Z i = {z i }}. Note that Z i also includes clutter measurements. And we define the subset w containing only one element to be a clutter subset. Thus, in this section, only the subset that is more likely generated by the target is obtained. By using Algorithm 1, we can obtain all subsets that satisfy the above conditions. In Algorithm 1, the inputs are the measurement sets Z i , the upper threshold D U , and the lower threshold D L . According to the distance partition method in [17], partitions of Z i are obtained (line 2 in Algorithm 1). The idea of the distance partition method is to group the adjacent measurements into the same subset. We can get different partitions based on different maximum partition distances d max ∈ [D L , D U ]. As the measurements generated by adjacent targets can also be allocated to the same subset, the subpartition algorithm [17] is applied (line 5 in Algorithm 1). In line 6, we delete the subset containing only one measurement. In the end, identical subsets are discarded to keep each subset unique.
Secondly, we apply the two-step partition method [10], [11] to obtain several partitioning hypotheses with higher weights. In the first step, the best associations for each Bernoulli component are obtained. As shown in Figure 1, the collection {w 1 i , . . . , w n i } calculated by Algorithm 1 is associated with the given j-th Bernoulli component (r  Step 1 of the two-step partition method.

FIGURE 2.
Step 2 of the two-step partition method.
After all the sensors are processed, the second step (as shown in Figure 2) is applied to select top P max partitions with the highest weight at most. Meanwhile, the Bernoulli components are processed sequentially. Note that there is no overlap between each measurement set W 1:s in the same partition.

V. SIMULATION EXPERIMENTS AND ANALYSIS
In this section, simulation experiments are conducted to verify the superiority of the proposed ET-MS-MeMBer filter. A comparison of the ET-MS-MeMBer filter and the IC-GGIW-CPHD filter is also given in this section.

A. PARAMETER SETTINGS
Consider a two-dimensional surveillance area of the size [−800, 800]m × [−800, 800]m, with three sensors and up to four targets observed in clutter. The detection probability of each sensor p i,d = 0.6, and the clutter intensity λ κ i = 5. Suppose that at time k, the target extension X (i) k is determined by where R (i) k is a rotation matrix that ensures the extension's major axis being aligned with the target's direction of motion. A i and a i are the length of the major and minor axes,  respectively. Here we assume that the number of measurements generated by the extended target is related to the target size: The parameters of the target are set as follows:     Figure 3. Figure 4 presents the measurements of sensor 1.
The parameters of the motion model are T s = 1s, θ = 1s, σ = 0.1, and τ = 5s. Set the newborn model according to the initial states of the target: M b,k = 3, r (1) b,k = r (2) b,k = r  ([11]). In Algorithm 1, we set D U = 60, and D L = 20. In the two-step measurement partition method, we set W max = 4 and P max = 4, i.e., the maximum number of subsets and hypotheses are both 4.
For a fair comparison, we adopt the same pruning threshold for both the ET-MS-MeMBer and the IC-GGIW-CPHD filter, and the threshold T cut = 10 −5 . The number of mixed components is no more than 100. In the IC-GGIW-CPHD filter, the merge thresholds of Gamma, Gaussian, and inverse Wishart components are set as U γ = 10, U G = 50, and U IW = 50, respectively.
We use the optimal subpattern assignment (OSPA) distance [31] as the metric for evaluating the filtering performance, in which the order c = 1, and the cutoff threshold p = 100. Given an estimated target state set G = {g 1 , · · · , g n } and a truth target state set Y = {y 1 , · · · , y m }(n ≤ m). The OSPA distance can be defined as   where m is the set of permutations on {1, · · · , m}. d (c) (g i , y π i ) = min(d(g i , y π i ), c), which denotes the distance between vector g i and y π i cutting off at a threshold c.

B. SIMULATION RESULTS
The filtering results of the ET-MS-MeMBer filter are shown in Figure 5. It can be seen that the proposed filter can effectively estimate the shape and position of the extended target. Figure 6 and Figure 7 compare the performance of the ET-MS-MeMBer filter and the MS-MeMBer filter over 100 Monte Carlo runs. It is clearly shown in Figure 6 that the MS-MeMBer filter has a significant bias in estimated cardinality, thus leading to a large OSPA distance in Figure 7. In contrast, the ET-MS-MeMBer filter can estimate the target cardinality more accurately and has a much smaller OSPA distance.
We also compare the performance of the ET-MS-MeMBer filter with the IC-GGIW-CPHD filter concerning different numbers of sensors. As before, 100 Monte Carlo runs are performed. Figure 8 and Figure 9 present the OSPA and cardinality estimation of the two filters with a varying number of sensors. The average OSPA and average running time  of the two filters are shown in Figure 10 and Figure 11, respectively. From Figure 9 and Figure 10, we can observe that the estimation accuracy of both filters improves as the number of sensors increases.
Meanwhile, compared with the IC-GGIW-CPHD filter, the ET-MS-MeMBer filter has a smaller average OSPA distance. From Figure 11, it can be seen that the average running time grows with the number of sensors. The reason for the increase is that more measurements need to be processed. By comparison, the average running time of the ET-MS-MeMBer filter is shorter than the IC-GGIW-CPHD filter. Therefore, we can conclude that the proposed ET-MS-MeMBer filter has a more satisfactory filtering performance with a lower computational cost.
In Figure 12 and Figure 13, we present the performance of the ET-MS-MeMBer filter and the IC-GGIW-CPHD filter with different clutter intensities. The monitoring area is observed by three sensors. We can see that with the increase of the clutter intensity, the average OSPA of the proposed ET-MS-MeMBer filter rises very slowly, while in contrast, the average OSPA of the IC-GGIW-CPHD filter grows strikingly and there is a steep growth between λ κ i = 5 and λ κ i = 65. In the meantime, in Figure 13, the average running time of the two filters show a similar trend as it is in Figure 12. Thus we can see that our proposed ET-MS-MeMBer filter has a superior performance in both filtering and efficiency.

VI. CONCLUSION
In this paper, we propose a novel ET-MS-MeMBer filter for extended target tracking to handle the bias of the estimated cardinality that arises in the existent MS-MeMBer filter. In addition, a GIW mixture implementation is introduced for the proposed ET-MS-MeMBer filter. We demonstrate that the proposed ET-MS-MeMBer filter can effectively estimate the shape and position of the extended target. Furthermore, the performance of the ET-MS-MeMBer filter is compared with the IC-GGIW-CPHD filter under different numbers of sensors, and simulations verify that the ET-MS-MeMBer filter outperforms the IC-GGIW-CPHD filter. It is also shown that the increasing number of sensors can lead to a higher filtering accuracy at the expense of more computational burden.

APPENDIX
In this section, we prove Propositions 1-3 given in Section III.B. Let G k|k [u] Under the APB model, the function φ g i (x) can be expressed as Thus, the derivative term in (A.3) can be written as   .
As the updated PGFL no longer has the multi-Bernoulli form in (7), we approximate it with a multi-Bernoulli component Therefore, we can obtain the updated multi-Bernoulli density according to (A.10).