Collaborative penalized Gaussian mixture PHD tracker for close target tracking

Signal Processing 102 (2014) 1–15

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Signal Processing journal homepage: www.elsevier.com/locate/sigpro

Collaborative penalized Gaussian mixture PHD tracker for close target tracking Yan Wang, Huadong Meng, Yimin Liu, Xiqin Wang Department of Electronic Engineering, Tsinghua University, Beijing, China

a r t i c l e i n f o

abstract

Article history: Received 6 August 2013 Received in revised form 28 November 2013 Accepted 14 January 2014 Available online 15 February 2014

Gaussian mixture probability hypothesis density (GM-PHD) recursion is a promising, computationally tractable implementation for the probability hypothesis density (PHD) filter. The competitive GM-PHD (CGM-PHD) and penalized GM-PHD (PGM-PHD) filters employ renormalization schemes to refine the weights assigned to each target and improve the estimation performance of the GM-PHD filter for closely spaced targets. However, these methods do not provide target trajectories over time, and the problem of wrongly identifying close targets is not still solved for the GM-PHD tracker. In this paper, we propose a collaborative penalized scheme to overcome the drawbacks of the GM-PHD tracker using the track label of each Gaussian component in the GM-PHD recursion. The simulation results show that the collaborative penalized GM-PHD (CPGM-PHD) tracker not only improves the estimation accuracy of the number of target and states but also provides the correct identities of targets in close proximity. & 2014 Published by Elsevier B.V.

Keywords: Multiple target tracking Probability hypothesis density Gaussian mixture PHD

1. Introduction Multi-target tracking (MTT) aims to jointly estimate an unknown number of targets as well as to predict their states and obtain target trajectories from a series of noisy measurements in the presence of spurious targets (clutter) and uncertainty of detection. The traditional methods based on data association, such as Global Nearest Neighbor (GNN), Joint Probabilistic Data Association (JPDA) [1–4] and Multiple Hypotheses Tracking (MHT) [1,2,5], all map the multi-target tracking into independent tracking of each single target by associating one measurement with one existing track at each time step. Using a random finite set (RFS) [6,7] to model the collections of targets and measurements as a whole, the finite set statistics theory (FISST) provides a rigorous Bayesian framework for MTT, and avoids the explicit associations between measurements and targets at the step of state estimation. The probability hypothesis density (PHD) filter [8,9] proposed by Mahler and realized by Vo et al. http://dx.doi.org/10.1016/j.sigpro.2014.01.034 0165-1684 & 2014 Published by Elsevier B.V.

propagates the first-order statistical moment or the intensity of the state RFS in time and is an effective approximation of the multi-target Bayesian posterior. There are two major implementations of the PHD filter: the particle-PHD filter (or SMC-PHD filter) [10–12] and the GM-PHD filter [13]. The latter propagates a sum of weighted Gaussian components representing the PHD function for linear Gaussian models. To obtain target trajectories, it is generally necessary to reprocess the identity-free estimates provided by the GM-PHD filter or the SMC-PHD filter. Some novel data association schemes for the SMC-PHD filter [14–16] are proposed to associate the target state estimates by reference to traditional association approaches such as MHT. The GMPHD tracker [17] determines the target trajectories directly from the evolution of the Gaussian mixture, where each single Gaussian demonstrates each possible trajectory [17–19]. When targets move near each other, such as in a small angle crossing or an occlusion condition, the performance of the original GM-PHD filter decreases dramatically. It is

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difficult for the GM-PHD tracker on its own to resolve the identities of targets in close proximity [17–20], or it leads to missing the estimates of the close targets [21–24]. The competitive GM-PHD (CGM-PHD) filter and penalized GMPHD (PGM-PHD) are proposed in the recent papers [22,23] and both employ renormalization schemes to re-manage the weights assigned to each target. When targets move near each other, the filters select the optimal measurement for each predicted target one-by-one by rearranging weights, which effectively overcomes the drawback of missing closely spaced targets for the GM-PHD filter. However, both filters do not provide continuous trajectories for the targets. The problem of wrongly identifying close targets for the GM-PHD tracker is not still solved effectively. For the CGM-PHD tracker and the PGM-PHD tracker implemented by arranging labels for Gaussian mixtures like the GM-PHD tracker, it is still difficult to distinguish between close targets in respectable scenarios. One possible outcome would be the assignment of the same identity to multiple target tracks, and another is that the identities are wrongly assigned to closely spaced targets. In MTT, there is such a one-to-one assumption that at each time step one measurement is generated from only one target and one target can only generate one measurement. In the association step of MHT, in a possible joint event of hypothesis trajectories, each trajectory is associated with a unique measurement series over time. The CGM-PHD tracker and the PGM-PHD tracker only consider the one-to-one association between each measurement at the current time step and each possible trajectory, but they ignore the unique history of the measurement series associated with each trajectory. Therefore, different measurements are possibly assigned to different trajectories but with the same identity for CGM-PHD or PGM-PHD, and targets are wrongly identified. Considering that there is a one-to-one correspondence between each measurement and each target with a different identity, in this paper we propose a novel scheme for weight rearrangement called collaborative penalized GM-PHD (CPGM-PHD) and utilize the track label of each Gaussian component in the GM-PHD recursion to collaboratively penalize the weights of targets with the same identity. Moreover, to avoid enlarging spurious targets by weight rearrangement we propose to process only the steady trajectories and leave the weights of the temporary targets unchanged. The scheme effectively reduces spurious trajectories generated by improper weight rearrangement in the CGM-PHD tracker and the PGM-PHD tracker. The performance of the proposed CPGM-PHD tracker is compared with the original GM-PHD tracker, the CGMPHD tracker and the PGM-PHD tracker in a number of experiments by simulating closely spaced targets with various uncertainties in noise, clutter rate and probability of detection. The simulation results show that our method not only improves the estimation accuracy of the number of target and states but also more accurately provides the identities of targets when they move near each other. The rest of the paper is organized as follows. Section 2 reviews the PHD filter and the GM-PHD tracker. Section 2 also analyzes the problem of wrongly identifying close

targets with the existing GM-PHD-based approaches. The proposed CPGM-PHD tracker is explained in Section 3. For illustration purposes, simulation results for our proposed method are given in Section 4. Finally, concluding remarks and possible future research directions are given in Section 5. 2. Problem formulation 2.1. Multi-target RFS modeling and the PHD filter In a multi-target system, considering that there is plenty of uncertainty in the target states and numbers because targets appear, dynamically move, split and disappear, multi-target states and measurements constitute RFSs. The RFSs of the multi-target states and measurements at time step k are modeled as X k ¼ fxk;1 ; …; xk;MðkÞ g and Z k ¼ fzk;1 ; …; zk;NðkÞ g, respectively. Thus, the MTT problem can be posed as a filtering problem in the RFS state space and measurement space. The PHD filter is an approximation of the multi-target Bayes filter, which makes the recursion computationally tractable. From the PHD function of the multi-target state RFS, the local maxima can be used to generate the state estimates of targets. The PHD recursion is as follows: Z Dk=k  1 ðxÞ ¼ pS;k ðζÞf k=k  1 ðxjζÞDk  1 ðζÞ dζ Z ð1Þ þ βk=k  1 ðxjζÞDk  1 ðζÞ dζ þ γ k ðxÞ;   Dk ðxÞ ¼ 1 pD;k ðxÞ Dk=k  1 ðxÞ þ ∑

z A Zk

pD;k ðxÞg k ðzjxÞDk=k  1 ðxÞ R ; κ k ðzÞ þ pD;k ðζÞg k ðzjζÞDk=k  1 ðζÞ dζ

ð2Þ

where γ k ðxÞ denotes the intensity of spontaneous target birth; f k=k  1 ðjÞ is the probability density of the Markov transition between target states; βk=k  1 ðjÞ denotes the intensity of a spawned target; g k ðjÞ is the likelihood function of measurement to state; κk ðzÞ is the intensity of the clutter RFS and equals λk ck ðzÞ, where λk is the mean number of clutter points in compliance with Poisson distribution; pS;k is the survival probability of targets and pD;k is the detection probability. For linear Gaussian models there is a closed-form solution for the PHD filter, called the GM-PHD filter. 2.2. GM-PHD filter and labeling Assuming that the Markov dynamic model according to which targets move and the measurement model are both linear Gaussian, i.e., f k=k  1 ðxjζÞ ¼ Nðx; F k  1 ζ; Q k  1 Þ;

ð3Þ

g k ðzjxÞ ¼ Nðz; H k x; Rk Þ;

ð4Þ

where Nðx; m; SÞ denotes a Gaussian density with mean m and covariance S; the parameter F k  1 is the state transition matrix; Q k  1 is the process noise covariance; Hk is the observation matrix and Rk is the observation noise covariance.

Y. Wang et al. / Signal Processing 102 (2014) 1–15

To reduce the computational load, pruning the Gaussian components with low weights and merging the Gaussian components within a certain distance are necessary.

Assuming that the intensities of the birth and spawn RFSs are both Gaussian mixtures, and the posterior intensity at time step k  1 is a Gaussian mixture, then, the predicted and updated PHDs are also Gaussian mixtures as follows: J k=k  1

Dk=k  1 ðxÞ ¼ ∑

j¼1

2.3. Drawback of the existing GM-PHD-based approaches wjk=k  1 Nðx; mjk=k  1 ; P jk=k  1 Þ;

ð5Þ

Ideally, in each tree structure representing each target, one rather large branch can be picked as an estimate. However, when targets move near each other, the situation is not so ideal. If the distances of more than one measurement towards one trajectory or multiple possible trajectories of one target are smaller than those towards other targets, more than one branch of an individual target would be reported as the estimated targets and some other target would be missed at that time. For this reason, the performance of the existing GM-PHD-based approaches decreases when tracking closely spaced targets. We take the example of tracking two closely spaced targets shown in Fig. 1. Regardless of merging, we show individually each Gaussian demonstrating each possible trajectory and multiple Gaussians in the Gaussian mixture are not summed. Suppose that there is no clutter and all of the measurements generated by the targets are detected. Two close targets have the identities A and B. Fig. 1 (1) shows the measurements at time step k 1 and the predictions of targets A and B. Because wBk  1=k  2 Nðz1k  1 ; zBk  1=k  2 ;

Dk ðxÞ ¼ ½1 pD;k ðxÞDk=k  1 ðxÞ þ ∑

J k=k  1

∑ wjk ðzÞNðx; mjk=k ðzÞ; P jk=k Þ

ð6Þ

z A Zk j ¼ 1

where J k=k  1 is the number of predicted Gaussian components; wjk=k  1 , mjk=k  1 and P jk=k  1 are the weight, mean and covariance of the j-th predicted component, respectively; mjk=k ðzÞ and P jk=k are the mean and covariance of the updated component by measurement z to the j-th predicted component, respectively. The GM-PHD tracker records the evolution of each Gaussian component. Assuming J k  1 is the number of Gaussian components at time step k 1, in the prediction process the same labels with the existing J k  1 Gaussian components are assigned to the corresponding predicted J k  1 components, and other spawn or birth components are newly labeled. According to (6), J k=k  1 predicted Gaussian components are kept in ½1  pD;k ðxÞDk=k  1 ðxÞ, and each measurement z A Z k is used to update all of the J k=k  1 -predicted Gaussian components. Each predicted component gives rise to 1 þ jZ k j Gaussian components with the same identity label, where jZ k j is the number of measurements. As a result, the updated Gaussian components for every predicted Gaussian component constitute a part of the tree structure. All branches of a tree have the same identity label and each branch is a possible trajectory of a target. The GM-PHD tracker is to select the branch with the largest weight as the individual target track from each tree demonstrating all possible tracks of each target.

P k  1=k  2 Þ is very small where zBk  1=k  2 is the predicted measurement of target B, there are only three updated components, shown as the curves A1, A2 and B1. Supposing that the means of A1, B1 and A2 are x1k  1 ; x2k  1 ; x3k  1 , respectively, the estimates x1k  1 and x2k  1 with weights w 40:5 are output. As shown in Fig. 1 (3), because 2 2 A1 wA1 k=k  1 Nðzk ; zk=k  1 ; P k=k  1 Þ is more than that of zk towards 1

other predicted measurements and zk is closer to the prediction of A1 than to other predictions, we obtain the four large updated Gaussian components shown in Fig. 1 (4). 0.2

A prediction B prediction

0.15

z1k−1: x=6.6

0.1

2

zk−1: x=2

0.05

−10

−5

0

5

0.2

0.05

2

−10

−5

0

5

x=2.5

10 (2) 1

0.2

2

z1: x=1.5 k 2 zk :

B1: B updated by zk−1

x (m)

PHD (k)

0.1

0.1

0

10

wk/k−1N(zk ;zA1 ,P ) k/k−1 k/k−1

A1 prediction A2 prediction B1 prediction

A2: A updated by z2k−1

(1)

A1

0.15

k−1

0.15

0.05

B 1 B N(z ;z ,P ) k−1/k−2 k−1 k−1/k−2 k−1/k−2

w

x (m)

PHD (k/k−1)

A1: A updated by z1

PHD (k−1)

PHD (k−1/k−2)

0.2

0

3

A11: A1 updated by zk

A12: A1 updated by z2 k 2

0.15

A21: A2 updated by zk 0.1

B11: B1 updated by z2 k

0.05 0

0 −10

−5

0

x (m)

5

10 (3)

−10

−5

0

x (m)

Fig. 1. An ambiguous example of tracking two close targets for the GM-PHD filter.

5

10 (4)

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Y. Wang et al. / Signal Processing 102 (2014) 1–15

There are two rather large branches of target A with weights w 4 0:5, and target B has only one large branch with weights far less than 0.5, as shown in the curve B11. Considering that the Gaussian components within a certain distance would be merged, the Gaussians A21 and B11 would be merged into one with only the identity A because the weight of A21 is bigger than that of B11. Then, the target B is missing. Regardless of merging, the means of A11 and A12 are reported as estimates. Let xn;m be the Gaussian mean by updating the prediction xnk=k  1 with the measurement zm, then, the means of A11 and A12, x1;1 and x1;2 , respectively, are reported. Assuming the weight of the branch xn;m is wn;m , Fig. 2(a) shows the tree structures of two targets in the example given in Fig. 1. The branches with the same color are generated at the same time step, and the height of the branch is equivalent to the weight of the branch. We briefly illustrate the principle of PGM-PHD and CGM-PHD using Fig. 2(b). After selecting the tallest branch (with the largest weight) of some tree, the other branches at the same node of the tree are cut short to make the branches of other trees grow by renormalization of the weights. After rearrangement by PGM-PHD, the new tree structures of the two targets are shown in Fig. 2(b), and the result of CGM-PHD is similar. The weaker tree B finally has a rather tall branch after the rearrangement. However, because only the branches at the same node with the tallest branch in the tree A are cut short, not only do the branches of the tree B grow but the branches from other nodes in the tree A also grow rapidly as shown in Fig. 2(b). Because the branch x3;2 in the tree A is also larger and taller than the biggest branch x2;2 in the tree B, the weaker tree B is still blocked and gradually withers over a period of time. So, for the PGM-PHD tracker and the CGMPHD tracker, one possible outcome would be the assignment of the same identity to multiple target trajectories, and the true identity of some target may be missing. Moreover, weeds (spurious target) may grow near the two trees.

Our goal is to make both trees representing two targets thrive as far as possible. Considering that there is a one-toone correspondence between each measurement and each target with a different identity, that is, some unique branch in each tree, we propose to cut short all of the other branches in the tallest tree while keeping the tallest branch to make the other tree nearby grow better. The ideal result is shown as Fig. 3. Our proposed method is called collaborative penalized GM-PHD tracker.

3. Collaborative penalized GM-PHD tracker This section explains the specific scheme of the collaborative penalized GM-PHD (CPGM-PHD) tracker. Let wj;m and w j;m be the weight and the normalized k k weight of the target xj;m respectively, k j j T ¼ pD;k wjk=k  1 Nðzm wj;m k ; H k mk=k  1 ; Rk þ H k P k=k  1 ðH k Þ Þ; k

ð7Þ

Fig. 3. Change in the tree structures of tracking two targets in the example given in Fig. 1 by the weight rearrangement of our proposed method.

Fig. 2. The tree structures of tracking two targets in the example given in Fig. 1. (a) Tree structure of GM-PHD. (b) Change of the tree structures for PGM-PHD.

Y. Wang et al. / Signal Processing 102 (2014) 1–15

w j;m ¼ k

wj;m k J

1 i;m κ k ðzm Þ þ ∑i k=k ¼ 1 wk k

n

;

ð8Þ

m

where zk is the m-th measurement at time step k. The parameters Rk and H k are defined in (4) and κk ðÞ is defined in (2). All of the unnormalized weights of the predicted targets constitute a J k=k  1  M k -order weight matrix M w , and the normalized weights construct another weight matrix M w , where M k is the number of measurements at time step k, and J k=k  1 is the number of Gaussian predicted components. The element 〈j; m〉 is the weight of the n o corresponding to the target xj;m ; P j;m that Gaussian mj;m k k k is created by updating the predicted target xjk=k  1 using the m

measurement zk . Each predicted target has an identity label

τjk=k  1 ,

and all of the Gaussians generated by the

same predicted target and different measurements have ¼ τjk=k  1 . Each Gaussian in the the same label, that is, τj;m k ; mj;m ; Gaussian mixture consists of four parts fwj;m k k P j;m ; τj;m g. k k When targets move near each other, and more than one measurement is closer to one target compared to the other targets, multiple Gaussians in the same row of the normalized matrix M w have large enough weights so that the j;m k total weight ∑M m ¼ 1 w k is greater than one. When Mk

41 ∑ w j;m k

5

ð9Þ

m¼1

the j-th target is called an inconsistent target. To overcome the drawback mentioned above, we propose to select the optimal association pair 〈jn ; mn 〉 between

target xjk=k  1 and all of the measurements according to the normalized matrix M w , and penalize all of the other n n Gaussians with the same label as τjk ;m by decreasing their weights in the unnormalized weight matrix Mw . Then, the weights in each column of the new M w are renormalized to construct a new normalized matrix. First, the target with the greatest weight in the normalized matrix M w is found as 〈jn ; mn 〉 ¼

arg max j A I and 8 m ¼ 1:M k

ðw j;m Þ; k

ð10Þ

where I ¼ f1; …; J k=k  1 g. In the algorithm of the original CGM-PHD filter or PGM-PHD filter, once the total weight of the remaining target with the maximum weight is less than one, the rearrangement process is terminated, sometimes improperly. Often the target with the largest weight meets the condition of target consistency and perhaps the other targets are still inconsistent according to (9). So we choose to check whether each target is inconsistent one by one until the total weight of each target in the remaining rows is less than one. To reduce the computational load, we can also set an end criterion, such as when the remaining maximum weight is less than a given nthreshold, e.g., 0.3. If n the remaining maximum weight w jk ;m is less than the given threshold, we have reason to believe that the total weight of the row is less than one. Even if this were not the case, the change by weight rearrangement would be n n slight because the penalization factor ð1  w jk ;m Þ is fairly large. n

j ;m k So, if the total weight ∑M is less than one, we m ¼ 1wk n ignore the j row and select the target with the greatest weight in the remaining rows until the total weight n

n

j;m k ∑M is greater than one. If w jk ;m is the greatest m ¼ 1wk

Fig. 4. Result of rearrangement by PGM-PHD and CGM-PHD. (1) Symbolic representation of unnormalized weights for the given example in Fig. 1. (2) Symbolic representation of normalized weights. (3) Unnormalized weight matrix before rearrangement. (4) Normalized weight matrix before rearrangement. (5) Result of rearrangement by PGM-PHD. (6) Result of rearrangement by CGM-PHD.

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Y. Wang et al. / Signal Processing 102 (2014) 1–15

weight, we penalize the other components except the element 〈jn ; mn 〉 in the jn row of the unnormalized weight matrix Mw and all of the components in the other rows n

n

with the same label as τjk ;m . The new unnormalized new weight matrix Mw is calculated as

wj;m ¼ k;new

8 j;m > w ; m ¼ mn > > k > n n > > < wj;m  ð1 w jk ;m Þ; k n

j ¼ jn ; 8 m ¼ 1 : M k ; ma mn n

> wj;m  ð1 w jk ;m Þ; ja jn ; τj;m ¼ τjk ;m ; 8 m ¼ 1 : M k ; ma mn > > k k > > > j;m n j;m jn ;mn : wk ; j a j ; τk a τk ; 8 m ¼ 1 : M k ; ma mn : n

n

ð11Þ new

Now, after penalization, a new normalized matrix M w will be constructed by w j;m ¼ k;new

wj;m k;new J

1 i;m κ k ðzm Þ þ ∑i k=k ¼ 1 wk;new k

:

ð12Þ

After one penalization and normalization, if the total n

j ;m k weight ∑M m ¼ 1 w k;new is still greater than one, then a new new

loop of penalization and normalization on the matrix Mw is necessary until the total weight

jn ;m k ∑M m ¼ 1 w k;new n n ð1  w jk ;m Þ.

is less than

one. The penalization factor is still Now we take the example given in Fig. 1 to illustrate the above procedure. Figs. 4 and 5 show the different values of weights due to the rearrangement of CGM-PHD, PGM-PHD and our CPGM-PHD. There are two predicted targets with the

same label A. As shown in Fig. 4(5) and (6), the target x3;2 achieves a higher weight than x1;2 ; x2;2 by the rearrangement of CGM-PHD or PGM-PHD, and the two filters report the targets x1;1 and x3;2 as their estimates. However, x1;1 and x3;2 have the same label A. Fig. 5 shows the numerical results after each step of processing for CPGM-PHD and the tracker finally reports the targets x1;1 and x2;2 as its estimates. The final results are obtained by two penalizations on target A and one penalization on target B. The intermediate result after one penalization and renormalization shown in Fig. 5(4) does not meet the condition of target consistency for W1R 41. It is necessary to penalize the weight matrix in Fig. 5(3) again. Finally, the sum of all of the weights in each row is less than one after renormalization as shown in Fig. 5(8). Generally, at most two penalizations for one target are enough. Multiple loops of penalizations are equal to one penan n lization by a factor of ð1  w jk ;m nÞn . We can simply explain j ;m k that the total weight ∑M m ¼ 1 w k;new is always reduced by penalization once again and each new normalized weight n jn ;m w k;new is no more than the nold w jk ;m . n Suppose that ρ ¼ 1 w jk ;m and the set of predicted n n targets with the same label as τjk ;m is S, then n

n

j ;m w k;new w jk ;m n

¼

ρwkj ;m κk ðzm Þ þ∑i 2= S wi;m þ ρ∑i A S wi;m k k k

Fig. 5. Numerical results after each step of processing for CPGM-PHD.

n



wjk ;m J

1 i;m κ k ðzm Þ þ ∑i k=k ¼ 1 wk k

Y. Wang et al. / Signal Processing 102 (2014) 1–15

7

n

¼

i;m ðρ 1Þwjk ;m ðκk ðzm = S wk ÞÞ k Þ þð∑i 2 J

1 i;m ðκ k ðzm Þ þ ∑i 2= S wi;m þ ρ∑i A S wi;m Þðκk ðzm Þ þ∑i k=k ¼ 1 wk Þ k k k k

o 0: n

;m is no more than So each new normalized weight w jk;new n

n

j ;m k the old w kj ;m , and the total weight ∑M m ¼ 1 w k;new is always

reduced by penalization and renormalization once again. Considering that some spuriously predicted targets might achieve large weights after renormalization to generate their spurious trajectories, we limit the rearrangement of weights on the steady trajectories. The estimates with the same label at multiple time steps construct a trajectory. If the number of dots in some trajectory is very small, i.e., the trajectory is short, we leave its original weights unchanged because we do not know whether the trajectory is spurious or true. Suppose that the label set of steady trajectories is Ts, final then the final weight matrix M w is recorded as

w j;m k;final

¼

8 < w j;m ; k;new : w j;m ; k

8 j ¼ 1 : J k=k  1 ; 8 m ¼ 1 : M k ; τj;m ATs k 8 j ¼ 1 : J k=k  1 ; 8 m ¼ 1 : M k ; τj;m 2 = Ts: k ð13Þ

If the label A of a new estimate belongs to the label set of the recorded trajectories, we add one to the number of tracking dots in the corresponding trajectory, otherwise we add the label A to the label set of the recorded trajectories and record that the trajectory A has one tracking dot. The label set of steady trajectories consists of the recorded trajectories that own more than a certain number of tracking dots, e.g., 5. Remark. In some special scenarios targets are so close that multiple Gaussian components representing different targets are merged into one. It is impossible to obtain estimates with different identities from such a merged Gaussian by any rearrangement of weights. In the original GM-PHD tracker, the same identity would be assigned to multiple tracks evolved from such a merged Gaussian at the next time step. It is also improper to penalize the updated targets that evolve from such a merged Gaussian in the CPGM-PHD tracker. Therefore, we propose to record each target that is merged from multiple Gaussians with different identities and skip the rearrangement of weights on such a target with multiple hidden identities. If the targets are too close, the problem of assigning the same identity to multiple tracks cannot be effectively solved by any GM-PHD-based approaches. However, we provide each track with hidden multiple identities with the hope of assisting other means of judging the true identity of each track, such as manual working.

Fig. 6. The overall scheme of the proposed CPGM-PHD.

The CPGM-PHD tracker is summarized by the pseudocode in Tables 1–3. 4. Simulation results

The overall scheme of our proposed CPGM-PHD tracker is illustrated as shown in Fig. 6. For τj;m ¼ τjk=k  1 ¼ τjk  1 , we k n

n

find the rows with the same label as τjk ;m n

τjk  1 ¼ τjk  1 .

when

For illustration purposes, we consider a Cartesian twodimensional target-tracking problem with a number of different probabilities of detection and clutter rates. The state xk ¼ ðpx;k ; p_ x;k ; py;k ; p_ y;k ÞT of each target consists of its

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Y. Wang et al. / Signal Processing 102 (2014) 1–15

Table 1 Pseudo-code for the CPGM-PHD tracker. Step 0. Initializations. Given X 0 ¼ fwj0 ; mj0 ; P j0 ; τj0 g and SI0 ¼ fI j0 g, j j 8 j ¼ 1 : J 0 . τ0 is identity tag. I0 is hidden identity set of the j-th target. m set k ¼1. Given Z k ¼ fzk g; 8 m ¼ 1 : M k . Step 1. Prediction Step 1.1 Prediction for born targets X γ;k=k  1 ¼ fwjr;k ; mjr;k ; P jr;k ; τjr;k g; 8 j ¼ 1 : J r;k . τjr;k are new tags. Step 1.2 Prediction for spawned targets X β;k=k  1 (details omitted) Step 1.3 Prediction for survived targets for j ¼ 1 : J k  1 wjS;k=k  1 ¼ pS;k wjk  1 ; mjS;k=k  1 ¼ F k  1 mjk  1 ; P jS;k=k  1 ¼ Q k  1 þ F k  1 P jk  1 F Tk  1 ; τjS;k=k  1 ¼ τjk  1 ; end

n o X S;k=k  1 ¼ wjS;k=k  1 ; mjS;k=k  1 ; P jS;k=k  1 ; τjS;k=k  1 ; 8 j ¼ 1 : J k  1 ; X k=k  1 ¼ X S;k=k  1 ⋃X γ;k=k  1 ⋃X β;k=k  1 ; Ijk=k  1 ¼ I jk  1 ; 8 j ¼ 1 : J k  1 ; I jk=k  1 ¼ τjk=k  1 ; 8 j 4 J k  1 ; Step 2. Construction of PHD update components for j ¼ 1 : J k=k  1 zjk=k  1 ¼ Hk mjk=k  1 ; Sjk ¼ Rk þ H k P jk=k  1 H Tk ; K jk ¼ P jk=k  1 HTk ðSjk Þ  1 ; end Step 3. Update Step 3.1. Update for undetected targets for j ¼ 1 : J k=k  1 wjk ¼ ð1  pD;k Þwjk=k  1 ; mjk ¼ mjk=k  1 ; P jk ¼ P jk=k  1 ; τjk ¼ τjk=k  1 ; Ijk ¼ I jk=k  1 ; end

n o X m;k ¼ wjk ; mjk ; P jk ; τjk ; 8 j ¼ 1 : J k=k  1 ; Step 3.2. Update for detected targets for m ¼ 1 : M k for j ¼ 1 : J k=k  1 j j wj;m ¼ pD;k wjk=k  1 Nðzm k ; zk=k  1 ; Sk Þ; k j mj;m ¼ mjk=k  1 þ K jk ðzm k  zk=k  1 Þ; k

¼ ðI  K jk Hk ÞP jk=k  1 ; τj;m ¼ τjk=k  1 ; P j;m k k end w j;m ¼ k

wj;m k J  1 j;m m κk ðzk Þ þ ∑j k=k ¼ 1 wk

; 8 j ¼ 1 : J k=k  1 ;

end w j;m ¼ w j;m ; 8 j ¼ 1 : J k=k  1 ; m ¼ 1 : M k ; k;old k

position ðpx;k ; py;k Þ and velocity ðp_ x;k ; p_ y;k Þ, where ½T denotes the transpose of a matrix ½. Each target moves according to the linear Gaussian model 2

1

60 6 xk ¼ 6 40 0

T

0

1

0

0 0

1 0

3

2

T 2 =2 6 7 6 07 T 7xk  1 þ 6 6 0 T5 4 1 0 0

0

3

7 0 7 7q k T 2 =2 7 5 T

ð14Þ

where qk is zero-mean Gaussian white noise with the variance Q ¼ diagð½s2v ; s2v Þ. Each target generates an observation according to  zk ¼

1

0

0

0

0

0

1

0

 xk þ r k

The pruning threshold T Th ¼ 10  5 , the merging threshold U¼4, the weight threshold wTh ¼ 0:5, and the maximum number of Gaussian components J max ¼ 100. In addition, the special parameters for the CPGM-PHD tracker include winTh ¼ 0:3 and U Th ¼ 1:5. In addition to checking the tracking results of four trackers in one simulation, we further study the performance of the trackers using three criteria: the accurate ratio of identifying every target (described later), the mean number of target estimation error (NTE) [13,22,23] and the mean optimal subpattern assignment (OSPA) [23,25] between the estimated target set X^ k and the true target set Xk, where NTEfX k ; X^ k g ¼ EfjX^ k j  jX k jg

ð15Þ

where rk is also zero-mean Gaussian white noise with the variance R ¼ diagð½s2ε ; s2ε Þ. The general parameters using the GM-PHD filter are as follows.

ð16Þ

  OSPAp;c X k ; X^ k " !#1=p jX k j 1 i ^ πðiÞ p p ^ ¼ min ∑ ðdc ðx ; x ÞÞ þ c  ðjX k j jX k jÞ jX^ k j π A Π jX^ j i ¼ 1 k

ð17Þ

Y. Wang et al. / Signal Processing 102 (2014) 1–15

9

Table 2 Pseudo-code for the CPGM-PHD tracker (continued from previous table). I ¼ f1; …; J k=k  1 g; While I a ∅ Step a. Find the target with maximum weight 〈jn ; mn 〉 ¼ n

arg max j A I and 8 m ¼ 1:Mk

ðw j;m Þ; k n

n

j ;m k if w jk ;m o win Th and ∑M r1 m ¼ 1wk break; end n

n

Step b. Find the targets with the same label as τjk ;m (equal to

n τjk  1 ) n

n

Sj ¼ fijτik  1 ¼ τjk  1 ; iA Ig; if

jn ;m k r1 ∑M m ¼ 1wk

n

n

or (wjk  1 4U Th and cardðIjk=k  1 Þ4 1)

jn

I ¼ I\fS g; else n

j ;m k While ∑M 41 m ¼ 1wk n

Step c. Penalize the weights of targets in Sj except the measurement mn n

n

ρ ¼ 1  w jk ;m ; for m ¼ 1 : M k if m a mn n

j wj;m ¼ wj;m nρ; 8 j A S ; k k end end Step d. Apply weight renormal- ization for m ¼ 1 : M k

w j;m ¼ k

wj;m k

J k=k  1

κk ðzm Þ þ ∑i ¼ 1 k

wi;m k

; 8 j ¼ 1 : J k=k  1 ;

end n

j ;m k 41) end (% end While ∑M m ¼ 1wk n

I ¼ I\fSj g; end end (% end While I a ∅ ) Step e. Keep the old weight of unsteady trajectories unchanged w j;m ¼ w j;m ; 8 j ¼ 1 : J k=k  1 ; 8 m ¼ 1 : M k ; τjk  1 2 = Ts; k k;old ; mj;m ; P j;m ; τj;m g; Ij;m ¼ Ijk=k  1 ; 8 j ¼ 1 : J k=k  1 ; 8 m ¼ 1 : M k ; X d;k ¼ fw j;m k k k k k X k ¼ X m;k ⋃X d;k ¼ fwik ; mik ; P ik ; τik g; SIk ¼ fI jk g; 8 i ¼ 1 : J k ;

where the parameters p and c define the order and the cut-off value of the OSPA metric and are set to 1 and 200, ^ ¼ minðc; dðx; xÞÞ ^ is respectively [23,25]. The distance dc ðx; xÞ the cut-off distance between the cut-off parameter c and ^ The set the Euclidean distance between the states x and x. Π jX^ k j represents the set of permutations of length jX^ k j with elements taken from f1; 2; …; jX^ k jg. The simulations are performed in the following three scenarios illuminated by different sensors to compare the performance of the proposed CPGM-PHD tracker with the original GM-PHD tracker, the CGM-PHD tracker and the PGM-PHD tracker. Because the CGM-PHD filter and PGMPHD filter do not provide the continuous trajectories of targets, we implement the two trackers by arranging labels to Gaussian mixtures such as the GM-PHD tracker. Scenario 1 is exactly the same as the scenario 1 in [22,23]. The sensor illuminating scenario 1 has a large measurement noise. In scenarios 2 and 3, the precision of the sensor is higher, but targets move more slowly and are closer. So trajectories may be incorrectly identified in scenarios 2 and 3. Scenario 1: The process noise and the measurement noise have covariances of Q ¼ diagð½0:2; 0:2Þ and R ¼ diagð½3200; 3200Þ, respectively. There are three targets moving in straight

paths in a region with size ½ 1000; 1000  ½0; 1000. The time step size T is set to 0.5 s. The targets enter the field of view (FOV) at time step 1, and exit from the FOV at time step 100. The probability of detection pD and the mean of clutter points λC are set to 0.99 and 2  10  6 m  2 , respectively. Figs. 7 and 8 show the detailed results for scenario 1. The true targets and the measurements generated by the targets are shown in Fig. 7(a). The tracking results of four trackers in one simulation are shown in Fig. 7(b). As shown in Fig. 7(b)(1), the original GM-PHD tracker loses some estimates of the target ‘1’ for a period when the three targets move near each other. Moreover, several estimates are a little off the true trajectory for target ‘2’. As shown in Fig. 7(b)(2) and (3), some spurious estimates or trajectories are generated by the PGM-PHD trackers and especially the CGM-PHD tracker. As shown in Fig. 7(b)(2), target ‘3’ is missing for a period and a new target ‘5’ is incorrectly generated in the place of target ‘3’. Fig. 7(b)(4) shows that our proposed approach provides the continuous and integrated trajectories of the three crossing targets. Fig. 8 demonstrates the corresponding results using NTE and OSPA in 100 runs. When the targets are far from each other, the error rates of the four trackers are nearly the same.

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Y. Wang et al. / Signal Processing 102 (2014) 1–15

Table 3 Pseudo-code for the CPGM-PHD tracker (continued from previous table). Step 4. Pruning and Merging Given a truncation threshold T Th , a merging threshold U, and a maximum allowable number of Gaussian components J max , set l ¼0, and I ¼ fi ¼ 1; …; J k jwik 4T Th g, repeat l ¼ l þ 1; j ¼ arg max wik ; iAI

L ¼ fiA Ijðmik  mjk ÞT ðP ik Þ  1 ðmik  mjk Þ r Ug; 1 ~ lk ¼ ∑ wik ; m ~ lk ¼ w ∑ w i mi ; ~ lk i A L k k w iAL h   i l 1 ~ lk  mik ðm ~ lk  mik ÞT ; τ~ lk ¼ τjk ; P~ k ¼ l ∑ wik P ik þ m ~ k iAL w Record the hidden identities in descending order of possibility I lk ¼ ∅; ½w; OL  ¼ sortðfwik jiA Lg; ‘descend’Þ; for i ¼ 1 : lenðOL Þ if wOL ðiÞ o 0:1 break; else I lk ¼ I lk ⋃IkOL ðiÞ ; end end I≔I\L; until I ¼ ∅. j j ~ jk ; m ~ jk ; P~ k ; τ~ jk g and fI~k g, 8 j ¼ 1 : l by those of the if l 4 J max replace fw j j ~ jk ; m ~ jk ; P~ k ; τ~ jk g and SIk ¼ fI~k g, 8 j ¼ 1 : l. J max Gaussian with largest weights. X k ¼ fw j ~ jk ; P~ k ; τ~ jk jw ~ jk 4wTh g, 8 j ¼ 1 : l. ~ jk ; m Output. fw Record the label set of steady trajectories (details omitted for lack of space).

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Fig. 7. The simulation results for the four trackers in scenario 1. (a) The true targets and measurements. (b) The simulation results.

For the clarity of the figure, we show the results of only the time steps 20–80. When the targets move near each other, NTE and OSPA are both much higher for the original GM-PHD trackers than for the improved trackers. The performance of the PGM-PHD and CGM-PHD trackers has improved significantly, while our proposed CPGM-PHD tracker has the lowest error rates throughout the simulation. Now, we study the performance of our method for tracking closer targets.

Scenario 2: In this scenario, there are three targets moving in straight paths in a region with size ½  500; 500  ½ 500; 500. They enter the FOV at time step 1, and exit from the FOV at time step 100. Scenario 3: In this scenario, the FOV also has size ½  500; 500  ½ 500; 500. Two targets enter the FOV at time step 1 and move towards each other until time step 75. After that, the targets move away from each other and exit from the FOV at time step 150.

Y. Wang et al. / Signal Processing 102 (2014) 1–15

For the two scenarios above, we set the process noise sv ¼ 1 m=s2 and the observation noise sε ¼ 16 m in all simulations. The time step size T is set to 1 s. Each target has a survival probability of ps;k ¼ 0:99 at each time step. Fig. 9(a) shows an example of the three crossing targets for scenario 2. Two of the three targets cross in turn. The three targets have velocities in the y-axis of v1y ¼ 8 m=s; v2y ¼ 0; v3y ¼ 8 m=s and all have the same velocity in the x-axis of vx ¼ 10 m=s. The probability of detection pD and the mean of clutter points λC are set to 0.99 and 10  10  6 m  2 , respectively. Fig. 9(b)(1) shows that for the GM-PHD tracker there are three target trajectories after the targets cross, but the same identity ‘2’ is assigned to two of the three trajectories. That is, the GM-PHD tracker cannot distinguish between target ‘2’ and target ‘3’ when they move near each other. Similar results are obtained by the PGM-PHD trackers shown in Fig. 9(b)(3). Fig. 9(b)(2) shows that the CGM-PHD tracker also cannot distinguish between the three targets, and the identities are wrongly assigned after

0.5

0 30

40

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70

80

time step GM−PHD CGM−PHD PGM−PHD CPGM−PHD

OSPA

80 60 40 20 20

30

40

50

60

70

80

time step Fig. 8. NTE and OSPA for the four trackers in scenario 1.

Remark. According to the tracking scheme of GM-PHD, we should choose the branch with the largest weight from the multiple branches owning the same label as the estimate of each target. However, because of the assignment of the same identity to multiple tracks when targets move near each other, there would be missing estimates if we choose only one from the multiple estimates with the same label. Therefore, we output all of the estimates with ~ k 4 wTh to more precisely estimate the number weights w of targets. We can still evaluate the estimation performance by analyzing NTE and OSPA. Fig. 10(a) shows another example of two closely spaced targets for scenario 3. The two targets have the same velocity in the x-axis of vx ¼ 6:7 m=s and initial velocities in the y-axis of v1y ¼ 10 m=s; v2y ¼  10 m=s. They move with constant velocity in the x-axis. In the y-axis, the two targets experience constant speed until time step 21, then constantly decelerate or accelerate until time step 129 with accelerations a1y ¼  0:185 m=s2 ; a2y ¼ 0:185 m=s2 , and then they experience constant speed until time step 150. The probability of detection pD and the mean of clutter points λC are set to 0.99 and 10  10  6 m  2 , respectively. Fig. 10(b)(1) shows that the GM-PHD tracker assigns the same identity ‘2’ to both targets after the two targets move close and then separate. The CGM-PHD tracker and the PGM-PHD tracker fail to correctly maintain the separate identities of target ‘1’ and target ‘2’, as shown in Fig. 10(b) (2) and (3), respectively. In addition, due to an improper weight adjustment, the CGM-PHD and PGM-PHD trackers generate a few spurious target estimates, and as a result, the trajectories of the targets fluctuate when the targets move close. original GM−PHD

true target & observation

300

3

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400

CGM−PHD

500

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20

the targets cross. Target ‘2’ is wrongly identified and the same identity of ‘1’ is assigned to two trajectories. Fig. 9(b)(4) shows that our proposed approach provides continuous and smooth trajectories with the three correctly identified crossing targets.

1

−400 −500 −500 −400 −300 −200 −100

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NTE

1

11

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400

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1

x(m)

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0

x (m)

Fig. 9. The simulation results for the four trackers in scenario 2. (a) The true targets and measurements. (b) The simulation results.

500

(4)

12

Y. Wang et al. / Signal Processing 102 (2014) 1–15

original GM−PHD true target & observation

500 2

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target 1 target 2 observation

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Fig. 10. The simulation results for the four trackers in scenario 3. (a) The true targets and measurements. (b) The simulation results.

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20 20

30

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time step

10 30

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time step

Fig. 11. NTE and OSPA for the four trackers. (a) Results for scenario 2. (b) Results for scenario 3.

Fig. 10(b)(4) shows the results of our proposed approach, which resolves the identities of the two targets without large fluctuations in the trajectories of targets. A Monte Carlo simulation with 100 runs is employed to evaluate the performance of the trackers using NTE and OSPA. Fig. 11(a) and (b) demonstrates the corresponding results of different trackers for scenarios 2 and 3. The probability of detection pD and the mean of the clutter points per unit area λC are set to 0.99 and 10  10  6 m  2 , respectively. For scenario 2, the performance of the PGMPHD and CGM-PHD trackers has improved greatly as shown in Fig. 11(a), but for scenario 3, both of the improved trackers make little improvement in the error rates as shown in Fig. 11(b). For the two scenarios, it can be observed that our proposed CPGM-PHD tracker has lower error rates than the other three trackers throughout the simulation.

To study the performance of our proposed method with higher uncertainties and to compare it with the other trackers, we provide the results of the four trackers for the two scenarios above with various clutter rates and probabilities of detection. For each configuration, a Monte Carlo simulation with 200 runs is employed to obtain reliable results. The metrics of NTE and OSPA are computed at each time step for 200 runs and then over time. Fig. 12(a) and (b) illustrates the average NTE and OSPA, respectively, for tracking targets in scenario 2. The horizontal axes in fig. 12(a) (1) and (b) (1) show the mean of the clutter points per unit area λC . Therefore, the number of clutter points per frame is equal to λC  V, where V ¼ 106 m2 is the area of the FOV for scenario 2. For different clutter rates, the probability of detection pD is set to 0.99, and for different probabilities of detection the clutter rate is fixed at λC ¼ 10  10  6 m  2 .

Y. Wang et al. / Signal Processing 102 (2014) 1–15

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Fig. 12. Averaged NTE and OSPA of tracking three targets (scenario 2) with different clutters and probabilities of detection. (a) Averaged NTE. (b) Averaged OSPA.

Fig. 12(a) (1) and (b) (1) shows that the CGM-PHD tracker and PGM-PHD tracker achieve similar error rates, which are lower than that of the original GM-PHD tracker. As shown in Fig. 12(a)(2) and (2), when the probability of detection decreases, the estimation error of all trackers tends to rise. For the same probability of detection, the CGM-PHD tracker and PGM-PHD tracker achieve almost the same error rates, resulting in an overlap of the curves of the two trackers. Our proposed CPGM-PHD tracker has the lowest error rates for all of the clutter rates and probabilities of detection shown in Fig. 12(a) and (b). Fig. 13(a) and (b) illustrates the average NTE and OSPA, respectively, for tracking two closely spaced targets in scenario 3. The configurations of pD and λC are the same as for scenario 2. Similar to the results of scenario 2, our proposed method has the lowest error rates in all of the simulations for scenario 3 shown in Fig. 13(a) and (b). In order of descending error rates, we have the original GMPHD tracker, the CGM-PHD tracker, the PGM-PHD tracker and the CPGM-PHD tracker. Next, we study the important performance of correctly identifying close targets for four trackers. The accurate rate for identifying every target is defined as the percentage of runs that the tracker correctly identifies and continuously tracks every target. For every simulation, we record the number of estimates in each trajectory with the same label. According to the features of the scenarios we studied, we propose to define the following two conditions that the tracking results need to satisfy before recognizing that the trackers correctly identify every target. (1) The number of long trajectories owning at least ðpD  n mo Þ tracking points is equal to the true number of targets where pD is probability of detection, n is the number of time steps (n ¼100 for scenario 2 and n¼150 for scenario 3) and mo is the given offset number (mo ¼ 25 for both scenarios). (2) The distance De between the first point and the last point in the long trajectory mentioned above is near that

of the true trajectory. Record the true distance Dt of the first point and the last point for target ‘1’. For scenario 2, there are two tracks with De more than ðDt 150Þ, and for scenario 3, there are two tracks with De less than ðDt þ 50Þ. Fig. 14(a) shows the accurate rates of tracking three targets for scenario 2 in terms of different clutter rates and probabilities of detection over 200 runs. As shown in Fig. 14(a), the CGM-PHD tracker and PGM-PHD tracker have substantially improved the performance of the GMPHD tracker in correctly identifying close targets, while our proposed approach has a higher accurate rate in all configurations and the accurate rates have been raised by at least 20 percent compared with the CGM-PHD tracker and the PGM-PHD tracker. Of course, when the probability of detection decreases, the accurate rate of our proposed approach also tends to decrease. However, the accurate rate is improved by at least 20 percent compared with the CGM-PHD and the PGM-PHD tracker in terms of different probabilities of detection. Fig. 14(b) shows the accurate rates of tracking two closely spaced targets for scenario 3 over 200 runs. When the clutter intensity is small, three improved trackers can identify each target with high accurate rates. However, when there is dense clutter, the performance of the CGMPHD and PGM-PHD trackers decreases dramatically. The CGM-PHD tracker results in an especially lower accurate rate than the original GM-PHD tracker. In all simulations, the CPGM-PHD tracker has the highest accurate rates for identifying close targets. For different clutter rates, the accurate rates of the CPGM-PHD tracker are greater than 90 percent. For different probabilities of detection, the accurate rates of the CPGM-PHD tracker are greater than 80 percent. When the probability of detection is relatively small, the perfect performance in correctly identifying close targets cannot be achieved. According to our simulation, when pD is 0.8, the accurate rate is just over 80 percent. In [13], it is discussed that the performance of the averaged NTE decreases sharply as pD decreases because the PHD filter has to resolve higher detection uncertainty

Y. Wang et al. / Signal Processing 102 (2014) 1–15

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1.5 (1) −5 x 10

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0.85

0.9

0.95 (2) 1

Probability Detection

Fig. 14. Accurate rate with different clutter rates and probabilities of detection: (a) results for scenario 2, (b) results for scenario 3.

on top of uncertainty in the number of targets. The problem of correctly tracking close targets in low detection of probability is more sophisticated, because the tracker has to resolve higher detection uncertainty in distinguishing between different targets. Our approach, based on rearranging the weights of inconsistent targets, cannot make effect due to a large loss of measurements. In order to compare the computation volume of the proposed method with the other three GM-PHD-based trackers, a statistical result is presented. The simulations are performed in tracking three close targets in scenario 2 for four trackers. The probability of detection pD and the mean of clutter points λC are set to 0.99 and 10 10  6 m  2 , respectively. The time complexity of the four methods based on GM-PHD is tested on a personal computer with Intel(R) Core(TM)2 Duo CPU 2.2 GHz, 2 GB RAM. The codes are run by MATLAB R2007B. For 500 Monte-Carlo trials, the averaged consumed time of PGM-PHD, CGM-PHD and CPGM-PHD trackers is 2.6356,

2.6112, and 2.6040 s, respectively, which are slightly larger than 2.5815 s, the counterpart of GM-PHD tracker. For CPGM-PHD tracker, it needs extra time to count up the steady trajectories and execute loops of penalizations before meeting the condition of target consistency, but it also reduces some computational loads by limiting the rearrangement of weight just on the steady trajectories and terminating the rearrangement process when the remaining maximum weight is less than a given threshold. So, in three improved methods our proposed CPGM-PHD tracker consumes minor computational efforts, which is close to the counterpart of the original GM-PHD tracker. 5. Conclusion To improve the performance of the GM-PHD-based filter for tracking closely space targets, a novel approach called collaborative penalized GM-PHD (CPGM-PHD) tracker is proposed. The proposed approach utilizes the one-to-one

Y. Wang et al. / Signal Processing 102 (2014) 1–15

correspondence between each measurement and each target with different identities to collaboratively penalize the weights of targets. By penalization and renormalization, the original weak targets are highlighted. The simulation results in terms of various probabilities of detection and clutter rates show that the proposed CPGM-PHD tracker has a better performance in identity confirmation than the original GM-PHD, CGM-PHD and PGM-PHD trackers, and the estimation accuracy of the number of targets and states is also improved. For future work, our research plan is to apply our approach to more scenarios where targets are closer, considering the separation distances of target estimates. References [1] Y. Bar-Shalom, T. Fortmann, Tracking and Data Association, Academic Press, Boston, 1988. [2] S. Blackman, R. Popoli, Design and Analysis of Modern Tracking Systems, Artech House, Boston, 1999. [3] K.-C. Chang, C.-Y. Chong, Y. Bar-Shalom, Joint probabilistic data association in distributed sensor networks, IEEE Trans. Autom. Control 31 (1986) 889–897. [4] T. Fortmann, Y. Bar-Shalom, M. Scheffe, Sonar tracking of multiple targets using joint probabilistic data association, IEEE J. Ocean. Eng. 8 (1983) 173–184. [5] S. Blackman, Multiple hypothesis tracking for multiple target tracking, IEEE Aerosp. Electron. Syst. Mag. 19 (2004) 5–18. [6] R. Mahler, Global integrated data fusion, in: Proceedings of Seventh National Symposium on Sensor Fusion, 1994, pp. 187–199. [7] I. Goodman, R. Mahler, H. Nguyen, Mathematics of Data Fusion, Kluwer Academic Publishers, Boston, 1997. [8] R. Mahler, Multitarget Bayes filtering via first-order multitarget moments, IEEE Trans. Aerosp. Electron. Syst. 39 (2003) 1152–1178. [9] R. Mahler, A theoretical foundation for the Stein–Winter probability hypothesis density (PHD) multi-target tracking approach, in: Proceedings of 2000 MSS National Symposium on Sensor and Data Fusion, vol. 1 (Unclassified), Sandia National Laboratories, San Antonio TX, 2000, pp. 99–117. [10] B.-N. Vo, S. Singh, A. Doucet, Sequential Monte Carlo methods for multi-target filtering with random finite sets, IEEE Trans. Aerosp. Electron. Syst. 41 (2005) 1224–1245.

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