Adaptive Collaborative Gaussian Mixture Probability Hypothesis Density Filter for Multi-Target Tracking

Transcription

1 sensors Article Adaptive Collaborative Gaussian Mixture Probability Hypothesis Density Filter for Multi-Target Tracing Feng Yang 1,2, Yongqi Wang 3, Hao Chen 1,2, Pengyan Zhang 1,2 and Yan Liang 1,2, * 1 School of Automation, Northwestern Polytechnical University, Xi an , China; yangfeng@nwpu.edu.cn (F.Y.); chenhao0727@mail.nwpu.edu.cn (H.C.); yybingxueer@163.com (P.Z.) 2 Key Laboratory of Information Fusion Technology, Ministry of China, Xi an , China 3 Southwest China Research Institute of Electronic Equipment (SWIEE), Chengdu , China; wangyongqi12@163.com * Correspondence: liangyan@nwpu.edu.cn; Tel.: Academic Editor: Xue-Bo Jin Received: 31 July 2016; Accepted: 3 October 2016; Published: 11 October 2016 Abstract: In this paper, an adaptive collaborative Gaussian Mixture Probability Hypothesis Density () filter is proposed for multi-target tracing automatic trac extraction. Based on the evolutionary difference between the persistent targets and the birth targets, the measurements are adaptively partitioned into two parts, persistent and birth measurement sets, for updating the persistent and birth target Probability Hypothesis Density, respectively. Furthermore, the collaboration mechanism of multiple probability hypothesis density (PHDs) is established, where tracs can be automatically extracted. Simulation results reveal that the proposed filter yields considerable computational savings in processing requirements and significant improvement in tracing accuracy. Keywords: multi-target tracing; multi-target state and trac extraction; GMPHD filter 1. Introduction In multi-target tracing (MTT) in clutters, the correspondence between the targets and the measurements is unnown, while the target number is unnown and even time-varying. The objective of MTT is to recursively estimate the target number and target states from a sequence of noisy and cluttered measurement sets [1,2]. One common approach for multi-target tracing is the combination of state estimation and data association, along trac initialization and termination [3]. In fact, data association and state estimation are coupled issues, i.e., the association ris triggers the measurement misuse and the estimation error increases the association ris. In other words, their direct combination above is in principle not suitable to the high uncertainty case; for example, dense targets/dense clutter [4]. Another approach is to apply random finite sets (RFSs) to represent the collection of individual targets and measurements, and hence recast the MTT problem as the Bayesian estimation problem based on finite set statistics so as to avoid data association ris [4]. However, the propagation of the multi-target posterior probability density function (PDF) is computationally intensive, which stems from the high-dimension integrations in multi-target state space. Mahler [5] proposed the first-order moment called the probability hypothesis density (PHD) of the PDF of the random set of state vectors. Vo and Ma [6] proved that the PHD surface is a Gaussian mixture (GM) in both the linear and Gaussian cases. Clar and Vo [7] analyzed the convergence property of the Gaussian Mixture Probability Hypothesis Density (GMPHD) filter. Up to now, PHD-related applications have been extended to many fields including visual target tracing [8], maneuvering target tracing [9,10], ground target tracing [11], extended target tracing [12,13], and sensor management [14]. Sensors 2016, 16, 1666; doi: /s

2 Sensors 2016, 16, of 16 However, there are still two important but open issues about PHD: The first is that the computational burden is still too intensive in many actual applications, especially in the case of dense clutters and intensive targets. One possible solution is to discern the measurement originality based on tracing gating [15,16]. Nevertheless, such a decision still faces mistae riss. The second issue is that additional trac extraction is needed because the standard PHD only outputs the trac points out the corresponding trac identities. But the presented trac extraction algorithm [17,18] is too complex to implement. In this paper, we present an adaptive collaborative GMPHD () filter the capability for automatic trac extraction for fast multi-target tracing in dense clutter. In the, the persistent and birth target PHDs are updated respectively based on the corresponding measurement subsets, instead of based on all measurements, so that the computational complexity is expected to be reduced. Meanwhile, the collaboration mechanism among these PHDs regarding measurement utilization is adaptively established to avoid decision riss triggered by measurement partitions. In addition, permanent tracs and temporary tracs can be automatically extracted from the persistent and birth target PHDs. Simulations show that the proposed greatly reduces the computational cost and significantly improves the trac extraction, compared the well-nown GMPHD filter. The remainder of this wor is organized as follows. Section 2 presents the MTT problem and briefly introduces the standard PHD filter. The is proposed in Section 3, and compared the GMPHD filter via simulation in Section 4. Finally, Section 5 concludes this paper. 2. Problem Formulation 2.1. Additional Gaussian Noise Model In an MTT scenario, targets appear and disappear randomly. A new target appears in the surveillance region either by spontaneous birth or by spawning from an existing target. The number of the newly-born targets is assumed to be Poisson-distribution. A target may disappear at the next instant. Here, p S, (x 1 ) represents the probability that a target corresponding to the state x 1 at 1 survives up to. For a target in the surveillance region, its movement is depicted by an additional Gaussian noise model: p 1 (x ζ) = N (x; f (ζ), Q 1 ) (1) define N ( ; m, P) is a Gaussian density function mean m and covariance P, so here, m = f (ξ), P = Q 1 ; One corresponding measurement will be obtained according to the following additional Gaussian noise model: p (z x) = N (z; h(x), R ) (2) where f is the state mapping for 1 to ; Q 1 is the covariance of process noise; h is the mapping from the state to the measurement; and R is the covariance of measurement noise. Denote x,i as the i-th target state, z,j as the j-th measurement, and N and M as the target number and measurement number, respectively. Now the multi-target states and observation sensor measurements are represented by the random finite sets (RFS) X = { x,1,..., x,n F (X ) and Z = { z,1,..., z,m F (Z), respectively. Here, F (X ) and F (Z) are the space of all finite subsets of state space X and measurement space Z, respectively. The RFS of the target states is as stated in [19]: and the RFS of sensor measurements is: X = S 1 (ζ) B (3) ζ X 1

3 Sensors 2016, 16, of 16 Z = Θ (x) C (4) x X where S 1 (ζ) is the survival target RFS inherited from the RFS X 1 ; B is the birth target ζ X 1 RFS; Θ (x) is the RFS of the detected target-originated measurements; and C is the RFS of clutter. x X 2.2. The PHD Filter In the PHD filter, the following common assumptions [6] are made: A.1: each target state evolves and generates one measurement independently; A.2: any clutter is Poisson distributed and independent of target-originated measurements; A.3: the predicted multi-target RFS is Poisson distributed and independent. Given the posterior intensity D 1 (x), the intensity function D 1 (x) is: and the posterior intensity D (x) is: D 1 γ 1 (x) + p s, (τ) f 1 (x τ)d 1 (τ)dτ (5) p D,(x)g (z x) D 1(x) D (1 p D, (x))d 1 (x) + z Z κ (z) + p D, (τ)g (z τ) D 1 (τ)dτ (6) where p D, (x) is the detection probability of an individual target state x; γ 1 ( ) is the birth target PHD at time ; κ ( ) is the intensity of the clutter RFS. Lemma 1. If the multi-target PHD at time 1 is represented as the sum of multiple PHDs each PHD having the following Gaussian mixture form [6]: then the predicted PHD will be where 1 1 D 1 1 D 1 1 N 1 1 (x) (7) ω (i,j) ( 1 N x; m (i,j) ) 1, P(i,j) 1 (8) N 1 (x) (9) 1 ω (i,j) ( 1 N x; m (i,j) ) 1, P(i,j) 1 and m (i,j) is the predicted mean; P(i,j) is the predicted covariance. 1 1 (10) ω (i,j) 1 = p S,ω (i,j) 1 (11) Lemma 2. Given the predicted PHD D 1 (x)and the measurements Z, the updated PHD [6] is: D N 1 (x) (12)

4 i1 i N1 N1 Sensors 2016, ( i) 16, 1666 ( i, j) ( i, j) ( i, j) ( i, j) ( i, j) ( i, j) W ( ) 4 of 16 x 1 pd, 1 x; m 1, P 1 ( z) x; m ( z), P j1 j1 zz i (13) where where (1 p D, ) ω (i,j) ( x; m (i,j) 1 p 1, D P(i,j), 1 ω (i,j) (z)n x; m ( z) (i,j) i z Z z; h( m ), N 1 N S 1 ( i, j) ( i, j) ( i, j) c( z) p z; h( m ), S 1 N ( ) ( (z), P (i,j) ) N 1 i N 1 i ( i, j) + ( i, j) ( i, j) 1 1 i1 j1 1 D, 1 1 (13) (14) ω (i,j) ω (i,j) 1 (z) = p D, ( c(z) + N 1 N 1 i ω (i,j) ( 1 p D,N z; h(m (i,j) ) N z; h(m (i,j) ) ), S(i,j) (14) Remar 1. As shown in the PHD recursions specified by Lemmas 1 2, the output 1 from the 1 PHD filter provides ), S(i,j) a multimodal density from which we need to estimate the states 1 of 1 the targets at each time, while it does not output target tracs. Moreover, the persistent target PHD and birth target PHD are jointly processed indifferently. Remar In fact, 1. Asthe shown birth in the PHD reflects recursions the specified nature by Lemmas of the birth 1 2, the target output and fromis the used PHDfor filter capturing provides a newborn targets while multimodal the persistent density fromphd whichis weupdated need to estimate for trac the states maintenance of the targetsof atthe eachsurviving time, while itargets. does not Utilizing output their target tracs. Moreover, the persistent target PHD and birth target PHD are jointly processed indifferently. difference is expected to further improve both tracing performance and calculation efficiency. Furthermore, all In fact, the birth PHD reflects the nature of the birth target and is used for capturing newborn targets while measurements (most of them are just clutter, for example in sea surveillance and ground tracing) are utilized the persistent PHD is updated for trac maintenance of the surviving targets. Utilizing their difference is to update expected the PHD, to further and hence improvemost both tracing computational performance resources and calculation are wasted. efficiency. These Furthermore, facts motivated all measurements us to establish the adaptive (mostmulti-phd of them are just filter clutter, framewor for example in seatrac surveillance extraction and ground and adaptive tracing) are processing utilized to update via both the adaptive measurement PHD, partition and hence most and multi-phd computationalcollaboration. resources are wasted. These facts motivated us to establish the adaptive multi-phd filter framewor trac extraction and adaptive processing via both adaptive measurement partition and multi-phd collaboration. 3. Adaptive Collaborative GMPHD () Filter Figure 3. Adaptive 1 is the Collaborative GMPHD framewor () consists Filter of persistent PHDs, birth PHDs, and prepersistent PHDs. Here, each PHD is specialized for a certain permanent/new-born/temporary target. In order to improve the computational efficiency, after the prediction of persistent PHDs and prepersistent PHDs, the measurements in the surveillance area are adaptively partitioned into the persistent measurements and the birth measurements. Furthermore, the persistent PHDs and the prepersistent PHDs are updated only using the persistent measurements, hence avoiding unnecessary data processing. Figure 1 is the framewor consists of persistent PHDs, birth PHDs, and pre-persistent PHDs. Here, each PHD is specialized for a certain permanent/new-born/temporary target. In order to improve the computational efficiency, after the prediction of persistent PHDs and pre-persistent PHDs, the measurements in the surveillance area are adaptively partitioned into the persistent measurements and the birth measurements. Furthermore, the persistent PHDs and the pre-persistent PHDs are updated only using the persistent measurements, hence avoiding unnecessary data processing. Measurements Measurement Partitioning Persistent Measurements Prediction of Persistent PHDs Prediction of Persistent PHD Prediction of Pre- Persistent PHDs Birth Measurements Update of Persistent PHDs Update of Persistent PHD Update of Pre- Persistent PHDs Efficient PHDs Invalid Measurements and PHDs Update of Birth PHDs PHDs Management Efficient PHDs Output Persistent PHD Prediction of Birth PHDs Birth Intensity Design Invalid Measurements Figure 1. Figure The adaptive 1. The adaptive collaborative Gaussian Mixture Probability Hypothesis Density Density () () filter framewor. filter framewor.

5 Sensors 2016, 16, of 16 Remar 2. One may wonder whether the measurement partition in the filter introduces several decision riss. In fact, the measurements away from the confidence region have little contribution to the persistent targets, and hence the weight of the Gaussian components would not be underestimated in updating the persistent PHD. In other words, the partition of the measurements has nothing to do updating the persistent targets in setting the large-confidence region. For the birth target, when the confidence region is large, the birth measurements may be possibly partitioned into the persistent measurements, which may lead to missing the birth target. Here the measurement which does not contribute to the update of the persistent PHDs will be introduced into the birth measurement set, which can avoid the birth target missing. In general, adaptive measurement partition is effective as shown in the later simulation Prediction of Persistent PHD Given the persistent PHD set {Ŵ (i) (i) p, 1, ˆl p, 1, ˆΞ (i) p, 1 N p, 1 and the pre-persistent PHD set {Ŵ (i) (i) b/p, 1, ˆl b/p, 1, ˆΞ (i) b/p, 1 N b/p, 1, the target intensity is represented in the Gaussian mixture form of multiple PHDs: N p, 1 D p 1 Ŵ (i) Ŵ (i) p, 1 p, 1 Nb/p, 1 p, 1 (x) + ˆω (i,j) ( p, 1 N x; ˆm (i,j) p, 1, ˆP (i,j) ) p, 1 Ŵ (i) b/p, 1 (x) (15) (16) Ŵ (i) b/p, 1 b/p, 1 ˆω (i,j) ( b/p, 1 N x; ˆm (i,j) b/p, 1, ˆP (i,j) ) b/p, 1 (17) where Ŵ (i) p, 1 (x) and Ŵ(i) b/p, 1 (x) denote the i-th persistent PHD and the i-th pre-persistent PHD; ˆl (i) (i) p, 1 and ˆl b/p, 1 are the corresponding labels; and ˆΞ (i) p, 1 and ˆΞ (i) b/p, 1 are the corresponding deleting thresholds, respectively. According to Lemma 1, the predicted persistent PHD is: N p, 1 D p 1 Nb/p, 1 p, 1 (x) + (x) (18) b/p, 1 where p, 1 b/p, 1 p, 1 p, 1 ω (i,j) ( p, 1 N x; m (i,j) ) p, 1, P(i,j) p, 1 ω (i,j) ( b/p, 1 N x; m (i,j) ) b/p, 1, P(i,j) b/p, 1 (19) (20) 3.2. Prediction of Birth PHDs According to reference [19], we choose the scheme of calculating birth PHD D b 1 : N b, 1 D 1 b = Ŵ (i) b, 1 (x) (21)

6 Sensors 2016, 16, of 16 Ŵ (i) b, 1 b, 1 ˆω (i,j) ( b, 1 N x; ˆm (i,j) b, 1, ˆP (i,j) ) b, 1 (22) where ˆω (i,j) b, 1, ˆm(i,j) b, 1, and ˆP (i,j) b, 1 are calculated based on the measurements which were not utilized for trac update at time 1. According to Lemma 1, the predicted birth PHD is: N b, 1 D b 1 = (x) (23) b, 1 where b, 1 p, 1 ω (i,j) ( b, 1 N x; m (i,j) ) b, 1, P(i,j) b, 1 (24) 3.3. Measurement Partition As the integral of the PHD equals the expectation of the target number in the surveillance region, the PHD should be updated by using the target-oriented measurements instead of all the measurements. Different inds of PHDs such as the persistent PHD and the birth PHD should be updated based on the different measurement sets. Thus, we partitioned the measurement space through utilizing the information of the Gaussian terms. The innovation covariances of the persistent and pre-persistent PHDs are: S (i,j) p, 1 = H P (i,j) p, 1 HT + R, s = 1,..., N p, 1, t = 1,..., N s p, 1 (25) S (s,t) b/p, 1 = H P (s,t) b/p, 1 HT + R, s = 1,..., N b, 1, t = 1,..., N s b, 1 (26) The shortest Mahalanobis distance between a real measurement z i Z and the predicted measurement is: d = min min i = 1,..., N p, 1 j = 1,..., N i p, 1 min s = 1,..., N b/p, 1 t = 1,..., N s b/p, 1 ( z i H m (i,j) ) T ( S (i,j) ) 1 ( z i p, 1 p, 1 H m (i,j) ), p, 1 ( z i H m (s,t) ) T ( S (s,t) ) 1 ( z i b/p, 1 b/p, 1 H m (s,t) ) b/p, 1 If d < τ α, then the measurement z i will be assigned to the persistent measurement set Zp Z, where the threshold τ α denotes the α quantile of the upper-tail of a chi-squared distribution n z degrees of freedom [1], where n z is the measurement dimension. The birth measurement set Z b Z is Z b = {z i Z z i / Zp (27) (28)

7 Sensors 2016, 16, of Update of Persistent PHD According to Lemma 2, the persistent PHD is updated based on the persistent measurement set Z p : N p, 1 D p p, b/p, 1 p D, + z Z p 1 p D, + z Z p L p, (z) = c p (z) + Nb/p, 1 p, (x) + (x) (29) b/p, p D, g (z x) (x) (30) L p, (z) p, 1 p D, g (z x) (x) (31) L p, (z) b/p, 1 p D, g(z x)d p (x)dx (32) 1 where c p (z) is the clutter intensity in the survival region; l (i) p. = l(i) p. 1 ; Ξ(i) p. = Ξ(i) p. 1 ; l(i) b/p. = l(i) b/p. 1 ; and Ξ (i) b/p. = Ξ(i) b/p. 1. The corresponding measurement weight of the persistent measurement is: Ψ p (z) = p D,g (z x)d p 1 (x) L p, (z) = N p, 1 Np, 1 i ω (i) Nb/p, 1 p, (z) + Nb/p, 1 i ω (i) (z) b, (33) The measurement set for updating the birth PHDs is defined as: { Z p/b = z Z p Ψp (z) < T z (34) where T z is the threshold for selecting the invalid measurements. Now the birth measurement set contains two parts: Z b = Zb p/b Z (35) The integrals of the persistent and pre-persistent PHDs are: ˆω (i) p, 1 p, = (1 p D, )ω (i,j) p, 1 p, 1 + z Z p ω (i,j) (z) (36) p, set by I b/p = ˆω (i) b/p, 1 b/p, = (1 p D, )ω (i,j) b/p, 1 b/p, 1 + z Z p ω (i,j) (z) (37) b/p, Denote the persistent PHD index set by I p = {1,..., N p, 1 and the pre-persistent PHD index {1,..., N b/p, 1. The invalid persistent and pre-persistent PHD index sets are: L p = L b/p = { i I P ˆω (i) p, < T 0 { i I b/p ˆω (i) b/p, < T 0 where T 0 is the upper-bound threshold of the weight of the invalid PHD. The invalid PHDs are treated as the additional birth PHDs in the following processing. (38) (39)

8 Sensors 2016, 16, of Update of Birth PHDs For any birth PHD, its prediction is: N b, 1 D b 1 b, 1 (x) + i L p p, 1 (x) + i L b/p N b, b/p, 1 (x) (40) b, 1 N b, = N b, 1 + L p + L b/p where L p represents the cardinality of the persistent targets; L b/p represents the cardinality of the invalid persistent targets. According to Lemma 2, the birth PHDs are updated using the birth measurements Z b : b, 1 p D, + z Z b N b, D b (x) (41) b, p D, g (z x) c b (z) + p D, g (z x) D b 1 (x)dx (x) (42) b, 1 where c b (z) is the clutter intensity in the birth region. Then the weight of the birth measurement and the integral of the birth PHD are calculated as: Ψ b (z) = p D,g (z x)d b 1 (x) L b, (z) (43) ˆω (i) b, 1 b, = (1 p D, )ω (i,j) b, 1 b, 1 + z Z p ω (i,j) (z) (44) b, and the invalid measurement set is constructed: Z,BI = {z Z b Ψb (z) < T z (45) Denote J b, = Z,BI as the number of birth PHDs at the next time step PHDs Management Birth PHDs Management If a birth PHD (x) has a large enough weight, such as ˆω(i) b, b, T 0, it can be reclassified as a { pre-persistent PHD. Then, the pre-persistent PHD set { b, (x) in the birth PHD set b, (x), i = 1,..., J b, Pre-Persistent PHDs Management W (j) b/p,. (x), j = 1,..., Nb/p 1 is augmented by If a pre-persistent PHD W (j) (x) has a large enough weight, such as ˆω(j) b/p, b/p, T 0, it can { be reclassified as a persistent PHD. Then, the persistent PHD set W (j) p, (x), j = 1,..., Np 1 is augmented by W (j) { b/p, (x) in the pre-persistent PHD set W (j) (x), j = 1,..., Nb/p b/p, 1.

9 Sensors 2016, 16, of Persistent PHDs Management A persistent PHD a large enough large weight, for example ω (i) p, T 0, is outputted as the stable trac. Otherwise, if ω (i) p, < T 0 up to successive E 0 time instants, then such a persistent PHD is considered as the terminated trac Birth Intensity Design Here we adopt the strategy previously described in reference [20] as follows. A one-step initialization method is utilized to select the reliable birth intensity components for the next time step. The measurements near the current multi-target states are deleted to reduce the unnecessary birth intensity components. Without velocity information, the a priori velocity is zero-mean and has the covariance determined based on the maximum expected velocity (For more details, see reference [21]). The invalid measurement set Z,BI is assigned to determine the birth PHD at time : D b J b, Ŵ (i) b, (x) (46) Ŵ (i) ( b, ˆω(i) b, N x; ˆm (i) b,, ˆP (i) ) b, (47) ˆm (i) b, = [z i, 0] T for z i Z,BI (48) [ ] ˆP (i) b, = R 0 0 VmaxI 2 M /3 (49) where the Gaussian term weight ˆω i b, is calculated according to reference [20]; I M is an M-by-M identity matrix; and M is the cardinality of the measurement set Z,BI Output Persistent PHD At different times, the Gaussian terms the same label represent the same target. The output trac information at time is the state ˆm (i) p, and covariance ˆP (i) p, of the persistent PHDs respectively: ˆm (i) p, = (i) xw p, (x)dx = (i) W p, (x)dx N i p, 1 (1 p D, )ω (i,j) ˆω (i) p, p, 1 m (i,j) p, 1 p, 1 + z Z p ω (i,j) p, (z) ˆω (i) p, m (i,j) (z) (50) p, ˆP (i) p, = = (i) (x ˆm p, )(x ˆm(i) p, )T p, (x)dx (i) W p, (x)dx N i p, 1 Np, 1 i + z Z p (1 p D, )ω (i,j) p, 1 ˆω (i) p, ω (i,j) p, (z) ˆω (i) p, [ P (i,j) p, 1 + (m(i,j) p, 1 ˆm(i) p, )(m(i,j) p, 1 ˆm(i) p, )T] [ ( ] P (i,j) p, + m (i,j) (z) ˆm(i) p, p, )(m(i,j) (z) ˆm(i) p, p, )T (51) Remar 3. The analysis for the complexity of the is an open issue due to the measurement partition is random. In principle, the and the standard GM-PHD have the similar process flowchart and hence their calculation burden is similar. Differing from the standard GM-PHD, the partitions all measurements into smaller sets and separately processes them. Since the computational complexity of the standard GM-PHD increases exponentially respect to the number of the related measurements, the is expected to be more cost-efficient, which coincides the simulation result.

10 Sensors 2016, 16, of Simulation Analysis To verify the systematic performance of the filter, we compared it the standard GMPHD filter which utilizes a priori birth target intensity and the GMPHD-I filter [20] via an MTT simulation scenario, similar to reference [8]. The difference is that six targets are considered here, instead of the three targets in reference [8]. The initial states, appearance, and disappearance of each target are given in Table 1. True trajectories are shown in Figure 2. Sensors 2016, 16, of 16 Table 1. A List of Initial Target States. Table 1. A List of Initial Target States. Target Target Index Index Appearing Appearing Time Time (s) (s) Disappearing Disappearing Time Time (s) (s) Initial Initial States States (m, (m, m, m, m/s, m/s, m/s) m/s) ( 1000, 500, 10, 10) ( 1000, 500, 5, 0) 0) (1050, 1070, 5, 5) (1050, 1070, 20, 5) ( 1000, 500, 0, 0, 20) 20) (1050, (1050, 1070, 10, 10) y/m clutter target 1 target 2 target 3 target 4 target 5 target x/m Figure Figure 2. Trajectories 2. of of the the true true targets and and the the clutters. The detailed scenario parameters are given in Table 2. The detailed scenario parameters are given in Table 2. Table 2. The Settings of the Simulation Scenario. Table 2. The Settings of the Simulation Scenario. Category Parameters Value Category sampling Parameters period Value 1 s region size (x-axis) [ 1500 m, 1500 m] sampling period 1 s region size (y-axis) [ 1500 m, 1500 m] region size (x-axis) [ 1500 m, 1500 m] clutter density m Scenario region size (y-axis) [ 1500 m, 1500 m] Scenario 2 2 sensor clutter noise density covariance λ R 4 diag(100 m 2 m,100 m ) sensor noise covariance R survival probability p diag(100 m 2, 100 m 2 ) s, 0.99 survival p s, 0.99 detection probabilityp Dp, D, state transition matrix F I2 I2 0 2 I 2 process noise standard deviation v 5 m/s I2 I process noise covariance Q

11 Sensors 2016, 16, of 16 Table 2. Cont. Category Parameters Value state transition matrix F [ I2 I 2 ] 0 2 I 2 process noise standard deviation σ v 5 m/s [ 2 4 process noise covariance Q σv 2 4 I 2 measurement matrix H [I ] maximum target speed v max 50 m/s initial birth Gaussian weight ˆω b, i 0.05 weight threshold T measurement weight threshold T z 0.1 PHD deleting threshold E I I 2 2 I 2 ] Priori birth target intensity γ 1 0.1N(x, x 1, P) + 0.1N(x, x 2, P γ ) is provided for the standard GMPHD filter x 1 = [ 1000 m, 500 m, 0 m/s, 0 m/s] T, x 2 = [1050 m, 1070 m, 0 m/s, 0 m/s] T, and P γ = diag {100, 100, 100, 100. We used a performance evaluation metric called the optimal subpattern assignment (OSPA) distance, which is specialized for the MTT filter accuracy test [22]. We selected OSPA parameters: p = 2 and c = 100. We ran 100 Monte Carlo (MC) trials for each filter to obtain the OSPA distance and the averaged computational cost, and obtained the trac-valued estimates in one MC trial. In addition, we validated the performance of each filter in the case of different clutter densities. Figures 3 and 4 show the Monte Carlo average of the OSPA distance detection probabilities of 0.9 and 0.7, respectively. Compared the other two filters, the almost always had the lowest OSPA distance, reflecting the effectiveness of the proposed adaptive multi-phd collaboration and measurement partition. As shown in Figures 3 and 4, there are five OSPA peas in the at times 20, 50, 60, 70, and 80 s, corresponding to the target appearances and disappearances, respectively. Some peas are even higher than that of the standard GMPHD, and are always smaller than that of the GMPHD-I. The explanation for this is that: the does not utilize a priori information of the birth target. At the moment that a target appears, the clutter and birth target measurements are hardly distinguishable out the support of the subsequent measurements. Thus, the birth target measurements may be treated as the clutter, and hence possibly leads to a delay in the cardinality estimation, as shown in Figure 5 when a target is newly born, which causes the pea of the OSPA distance. This is the cost of the measurement partition. the GMPHD-I also does not utilize a priori information of the birth target. However, due to the absence of adaptation and collaboration compared the, the GMPHD-I is the worst regarding the OSPA measure. Furthermore, we present the time-averaged OSPA versus the clutter density in Figure 6. With the increase of clutter density, the OSPA distance of the filter gradually increases, but it is still lower than that of other two comparison algorithms. For each time step, the averaged computation time (ACT) is shown in Figure 7. Obviously, the filter significantly decreased the computational burden, compared the standard PHD or GMPHD-I filters. The ACT of the filter is about 0.65 s, much smaller than the measurement sampling period of 1 s.

12 Sensors 2016, 16, of 16 Figure 8 plots the curves of the ACT versus the clutter density. As the clutter density increases, Sensors 2016, 16, of 16 the ACTs of all the filters increase; however, the filter has the lowest rate of increase, Sensors 2016, 16, of 16 implying that it is more suitable for the dense clutter case GMPHD-1 GMPHD GMPHD-1 GMPHD OSPA OSPA time/s time/s Figure 3. Monte Carlo average of the OSPA distance a detection probability of 0.9. Figure 3. Monte Carlo average of the OSPA distance a detection probability of 0.9. Figure 3. Monte Carlo average of the OSPA distance detection probability of GMPHD-1 GMPHD GMPHD-1 GMPHD OSPA OSPA time/s time/s Figure 4. Monte Carlo average of the OSPA distance a detection probability of 0.7. Figure 4. Monte Carlo average of the OSPA distance a detection probability of 0.7. Figure 4. Monte Carlo average of the OSPA distance a detection probability of 0.7.

13 Sensors 2016, 16, of 16 Sensors 2016, 16, of 16 Figure 5. Monte Carlo average estimates of the number of targets. Estimated number (solid lines), Figure standard 5. Monte deviation Carlo (dashed average lines). estimates of the number of targets. Estimated number (solid lines), standard deviation (dashed lines) GMPHD-I GMPHD-I time-average time-average OSPA OSPA clutter 3density 4 5 x clutter density x 10-6 Figure 6. Time-averaged OSPA distance in one run of each filter for varying clutter densities. Figure 6. Time-averaged OSPA distance in one run of each filter for varying clutter densities. Figure 6. Time-averaged OSPA distance in one run of each filter for varying clutter densities.

14 Sensors 2016, 16, of 16 Sensors 2016, 16, of 16 Sensors 2016, 16, of computation computation time/s time/s GMPHD-I GMPHD GMPHD-I GMPHD time/s time/s 1.4 Figure 7. The computation time of the three filters. Figure 7. The computation time of the three filters. Figure 7. The computation time of the three filters GMPHD-I GMPHD GMPHD-I GMPHD 1 1 computation computation time/s time/s clutter density 0 x clutter density x 10-6 Figure 8. The time-averaged computation time under different clutter densities. Figure 8. The time-averaged computation time under different clutter densities. 5. Conclusions 5. Conclusions In this paper, we proposed the filter automatic trac extraction, which was shown In to this be be paper, satisfactory we proposed for for multi-target the tracing. It costs filter It costs less computational less automatic trac timextraction, and has and better which has OSPA, better was except shown OSPA, for except to some be satisfactory for undesirable some undesirable for peas multi-target atpeas the moment at tracing. the moment of birth It costs of targets birth less appear. computational targets appear. Hence, one time Hence, possible and one has possible avenue better for OSPA, avenue future except for research future for some research is toundesirable introduce is introduce priori peas information priori the moment information about of birth about targets, birth appear. which targets, would Hence, which be one helpful would possible be in order avenue helpful toin for better order future discern to research better the target discern to and introduce the the target clutter, priori and and the information hence clutter, more and about effectively hence birth more reduce targets, effectively the which corresponding reduce would the be helpful in order to better discern the target and the clutter, and hence more effectively reduce the

15 Sensors 2016, 16, of 16 OSPA peas. Additionally, the threshold at the step of PHDs management was considered as constant in the proposed filter, and a possible future research study is to adaptively choose the threshold under some optimal performance index. Acnowledgments: The authors would lie to acnowledge the support of National Natural Science fund of China, No , and Author Contributions: Feng Yang and Yongqi Wang conceived of, designed and performed the simulations and wrote the manuscript. Hao Chen, Pengyan Zhang and Yan Liang reviewed the manuscript. Conflicts of Interest: The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results. References 1. Blacman, S.S.; Popoli, R. Design and Analysis of Modern Tracing Systems; Artech House: Norwood, MA, USA, Bar-Shalom, Y.; Li, X.R. Multitarget Multisensor Tracing: Principles and Techniques; YBS Publishing: Storrs, CT, USA, Bar-Shalom, Y. Tracing and Data Association; Academic Press: Orlando, FL, USA, Mahler, R.P.S. Statistical Multisource-Multitarget Information Fusion; Artech House, Library: Norwood, MA, USA, Mahler, R.P.S. Multitarget Bayes filtering via first-order multitarget moments. IEEE Trans. Aerosp. Electron. Syst. 2003, 39, [CrossRef] 6. Vo, B.N.; Ma, W.K. The Gaussian mixture probability hypothesis density filter. IEEE Trans. Signal Process. 2006, 54, [CrossRef] 7. Clar, D.; Vo, B.N. Convergence analysis of the Gaussian mixture PHD filter. IEEE Trans. Signal Process. 2007, 55, [CrossRef] 8. Pollard, E.; Plyer, A.; Pannetier, B.; Champagnat, F.; Besnerais, G.L. GM-PHD filters for multi-object tracing in uncalibrated aerial videos. In Proceedings of the 12th International Conference on Information Fusion, Seattle, WA, USA, 6 9 July Zhang, Q.; Song, T.L. Improved Bearings-Only Multi-Target Tracing GM-PHD Filtering. Sensors 2016, 16, [CrossRef] [PubMed] 10. Dong, P.; Jing, Z.L.; Li, M.Z.; Pan, H. The variable structure multiple model GM-PHD filter based on liely-model set algorithm. In Proceedings of the 19th International Conference on Information Fusion, Heidelberg, Germany, 5 8 July 2016; pp Wu, W.H.; Liu, W.J.; Jiang, J.; Gao, L.; Wei, Q.; Liu, C.Y. GM-PHD filter-based multi-target tracing in the presence of Doppler blind zone. Digit Signal Process 2016, 52, [CrossRef] 12. Dai, B.; Li, C.Y.; Ji, H.B.; Hu, Y. Iterative-Mapping PHD filter for extended targets tracing. In Proceedings of the 2015 International Conference on Control. Automation and Information Sciences, Changshu, China, October 2015; pp Granstrom, K.; Orguner, U. A PHD filter for tracing multiple extended targets using random matrices. IEEE Trans. Signal Process. 2012, 60, [CrossRef] 14. Mahler, R.P.S.; Elfallah, A. Unified sensor management in unnown dynamic clutter. Soc. Photo-Opt. Instrum. Eng. SPIE Process. 2010, 7698, Macagnano, D.; De-Abreu, G.T.F. Adaptive gating for multitarget tracing Gaussian mixture filters. IEEE Trans. Signal Process. 2012, 60, [CrossRef] 16. Li, T.; Sun, S.D.; Sattar, T.P. High-speed Sigma-gating SMC-PHD filter. Signal Process. 2013, 93, [CrossRef] 17. Li, Y.X.; Xiao, H.; Song, Z.Y.; Fan, H.Q.; Hu, R. Joint multi-target filtering and trac maintenance using improved labeled particle PHD filter. In Proceedings of the IEEE International Congress on Image and Signal Processing (CISP), Datong, China, October 2014; pp Panta, K.; Vo, B.N.; Singh, S. Novel data association schemes for the probability hypothesis density filter. IEEE Trans. Aerosp. Electron. Syst. 2007, 43, [CrossRef]

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