1. Introduction
Multi-target tracking (MTT) has to deal with the detection and estimation problems of multiple targets in a cluttered environment [
1]. Traditional solutions such as multiple hypothesis tracking (MHT) filter and joint probabilistic data association (JPDA) filter handle this problem through a divide-and-conquer approach that involves data association and filtering processes [
2,
3]. As an alternative to the association-based methods, the random finite sets (RFS) approach is an emerging technique to multi-target tracking (MTT), and the resulting optimal multi-target Bayes filter provides a rigorous theoretical basis for many novel multi-target filters [
4,
5,
6]. In this context, the probability hypothesis density (PHD) filter [
4] which is derived via first-order moment approximation of the multi-target posterior density, and its implementations such as sequential Monte Carlo PHD (SMC-PHD) filter [
7,
8] and Gaussian mixture PHD (GM-PHD) filter [
9], have been widely studied in the area of MTT over the last decade [
10,
11,
12,
13,
14].
From an engineering point of view, the SMC-PHD filter is more suitable for practical applications due to its ability to accommodate both nonlinear and non-Gaussian dynamics [
8,
13]. However, the SMC method leads to a troublesome problem in extracting the estimates of target states from the given particle approximation of the PHD (also known as intensity function), and the accuracy of the estimated multi-target state directly determines the tracking performance of a multi-target filtering algorithm. Thus it is a critical issue for the SMC-PHD filter to develop a reliable multi-target state extraction algorithm, which has attracted significant attention [
15,
16,
17,
18,
19,
20,
21]. Typically, the clustering techniques such as k-means clustering [
7,
15] and c-means fuzzy clustering [
16], and finite mixture models (FMM) algorithm via expectation-maximization (EM) [
15] or Markov chain Monte Carlo (MCMC) sampling [
17], have been investigated in this aspect. The results in [
15] demonstrated that the k-means clustering outperforms the FMM algorithm with potential computational efficiency and fewer spurious estimates. In addition, the CLEAN method [
18] was proposed to extract target states from the SMC-PHD filter. However, since this method only exploits the weight information of particles, the average performance of the CLEAN method is no better than the
k-means clustering in general. Subsequently, the method in [
19] introduced clustering algorithms to overcome the drawbacks of the CLEAN technique. Despite this, the effect is limited because the clustering algorithms [
15,
16] are particularly problematic and unreliable due to the hard limiting on the number of clusters (specified by the estimated number of targets), especially in the scenarios where there exist closely spaced targets or incorrect estimates of target number [
19,
20,
21]. Moreover, such methods have high computational cost because of the requirement of iterative computing process.
To address these problems, some measurement-oriented methods that consider the relationships between weighted particles and measurements have been proposed to perform state extraction [
20,
21], where the
ad-hoc particle clustering methods mentioned above are eliminated. Typically, the grouping method [
20], implemented by replicating the current particle set and re-weighting them corresponding to individual measurements, has been a popular state extraction method in different versions of the SMC-PHD filter [
10,
12]. In contrast, the method in [
21] introduced a maximum likelihood criterion for particle clustering. More recently, a systematic and theoretical analysis about the possibility of extracting point estimates from PHD with respect to the optimal sub-pattern assignment (OSPA) [
22] metric was investigated in [
23], however, how to design a reasonable loss function and extend the method to general cases need to be further researched. To facilitate the parallel implementation of the SMC-PHD filter, a new multi-expected a posterior (MEAP) method was outlined in [
24], where the estimation procedure can still be seen as a kind of measurement-oriented technique by introducing particle-to-measurement association based on the near and nearest neighbor principle. Moreover, the concept of association measure was introduced to the PHD recursion [
25], which theoretically makes it possible to extract track estimates from the PHD filter. Unfortunately, due to the complexity of measurement association and the fast augmentation of the number of observation paths, the feasible implementation method and its potential performance are not clear. Generally speaking, the measurement-oriented methods are more acceptable and practical than the clustering-based methods in terms of both reliability and computational efficiency [
20,
24]. On the other hand, the existing solutions of this category are all limited by the fact that only the targets that have been detected may have chances to be reported, which will result in inaccurate estimation when missed detections occur. Since the PHD recursion is sensitive to missed detections, to date, how to extract state estimates from the SMC-PHD filter accurately in the presence of detection uncertainty still remains a challenge.
The key contribution of this paper is a novel solution to achieving multi-target state extraction in the SMC-PHD filter. More specifically, the normalized likelihoods of the predicted particles with respect to individual measurements are introduced to develop a validation mechanism which aims at selecting effective measurements and particles, i.e., the target-originated measurements and the particles corresponding to detected targets. Subsequently, by constructing the association probability distributions between particles and measurements, the particles of detected targets are divided into different clusters corresponding to the effective measurements in a probabilistic manner. Then, according to the estimated target number, the point estimates of the target locations can be extracted from the resulting clusters. Moreover, benefiting from the proposed validation mechanism, we introduce a gating technique to further identify the particles of undetected targets, and then extract the corresponding target states, thereby implying an improved estimation performance in the circumstances with detection uncertainty. Simulation results demonstrate the effectiveness of our methods in comparison with the existing methods.
The remainder of this paper is organized as follows:
Section 2 provides a brief review of the PHD filter and the SMC-PHD filter. The proposed multi-target state extraction method is presented in
Section 3. Simulation results and analysis are presented in
Section 4 and conclusions are drawn in
Section 5.
4. Simulation
To evaluate the performance of the proposed methods, a two-dimensional tracking scenario with an unknown and time-varying number of targets is considered, where the target dynamic model is exactly the same as that in [
7,
24]. The position of the sensor platform is assumed to be known at coordinate origin, and the observation equations are given by:
where
εk,1 and
εk,1 are the zero-mean Gaussian white noise with respective standard deviations π/180° and 2 m. Assuming that the survival probability of each target is independent of its state and the value is set to be
pS,k = 0.98 during the simulations. Clutter is uniformly distributed over the region [0, π/2](rad) × [0,700] (m), and the number of clutter measurements per scan is Poisson distributed with a specified mean value
r.
β = 500 is used in the SMC-PHD filter, and thus the number of particles varies according to the estimated number of targets. The parameters
γg = 1/
β ,
τn = 100 and λ= 25 (corresponding to
pg = 0.9999) are used for the proposed Algorithms 1 and 2. Besides, the OSPA metric [
22] is adopted to evaluate the estimation performance of different methods. The intensity of newborn targets is modelled as [
7,
15,
24]:
where N(
x;
m,P) denotes a normal distribution with mean
m and covariance
P, and the values of
and
are configured as
,
,
and
. We compare the performance of the proposed methods with that of the
k-means method [
15], grouping method [
20] and MEAP method [
24] via Monte Carlo (MC) simulations.
In the surveillance region, we design a relatively complex multi-target environment with crossing tracks and paralleling motion in close range. More specifically, target 1 exists in the surveillance region from time step 1 to 30. Target 2 and target 5 appear at time step 20 from different positions and their track crossing happens at time step 45. Target 3 and target 4 appear at time step 15 simultaneously and they keep parallel motion until time step 50.
Figure 1 shows the true tracks of 5 targets in this scenario. To capture the localization errors of different state extraction methods, the parameters
p = 2 and
c = 20 are chosen to generate OSPA metric value.
To verify the effectiveness of the proposed Algorithms 1 and 2 in an intuitive manner, we present a typical example of the filter output at time step 35 (missed detection occurs). The probability of detection and clutter rate are set to be
pD = 0.90 and
r = 10 in the simulation, respectively. There exist four true targets during this iteration, while the estimated number of targets is
N35 = 3.
Figure 2 gives the global distribution of the predicted particles corresponding to persistent targets at this time step. Besides, the true locations of targets and the extracted locations using Algorithm 1 are also displayed in the same figure. It is clear that the proposed Algorithm 1 provides accurate estimation of the existing three targets based on the estimated number of targets, although the particles of the target 3 and target 4 exhibit significant overlap due to the parallel motion in close space. Besides, target 2 is an undetected target and the existing measurement-oriented methods cannot extract its state in this case, including the proposed Algorithm 1.
Subsequently, the proposed Algorithm 2 is performed.
Figure 3 shows the particles (in U
k) corresponding to undetected targets at this time step, as well as the extracted target state from these particles. The results demonstrate that the proposed Algorithm 2 can effectively identify the potential particles of undetected target and then extract the state estimate. As shown in
Figure 3, there are a few spurious particles around the birth regions. This phenomenon is caused by the resampling process in the previous iteration, where the filter needs to draw particles from the birth intensity to exploit potential new targets at each scan.
To compare the average performance, 200 MC runs are performed for the SMC-PHD filter with different state extraction methods. The target tracks are fixed but clutter and measurements are independently generated for each trial. Note that the effectiveness of the proposed Algorithm 2 is examined by combining its estimated results with that of the Algorithm 1, namely, the modified results in Equations (20) and (21), in subsequent simulations.
Figure 4 shows the statistical results of the estimated number of targets and mean OSPA distances of five methods at each time step. The true number of targets and the estimated results from SMC-PHD filter are shown in
Figure 4a. Based on the extracted multi-target state using different methods, the corresponding mean OSPA distances are shown in
Figure 4b.
As shown in
Figure 4a, the standard SMC-PHD filter gives a satisfactory performance on the target number estimation in this scenario, while the modified estimate by Algorithm 2 follows the true value more closely. In terms of state estimation, it can be seen from
Figure 4b that the grouping method shows a slight advantage over the k-means clustering method at most time steps but has a large error when new targets appear. The reason is that the method only reports the estimates whose total weights are above a certain threshold value (we adopt 0.8 as was done in [
20]), which is not favorable for the estimation of newborn targets. By contrast, the superiorities of the MEAP method and the proposed methods are remarkable. It can be seen that the proposed Algorithm 1 exhibits a better performance than that of the MEAP method. This can be attributed to the proposed mechanism for selecting particles and measurements, where the contributions of particle likelihoods are evaluated and the scopes of particles are restricted corresponding to effective measurements. Such kind of particles can exactly capture the regions of PHD peaks for state extraction. Moreover, the combination of Algorithms 1 and 2 achieves the best estimation accuracy as compared with all other methods in the presence of detection uncertainty.
Figure 4b also indicates that the estimation of undetected targets will cause a short delay response (exhibits large error at time steps 31 and 51) when targets really disappear. In reality, it is difficult to discriminate the two cases in a single iteration.
To present a more comprehensive evaluation, we also study algorithm performance against different detection probabilities via MC simulations with fixed clutter rate
r = 10. In addition, similar simulations are performed against various clutter rates under the condition of a constant detection probability
pD = 0.90. We compute the time-averaged OSPA distance of each method during these simulations. The results are presented in
Figure 5 and
Figure 6, respectively.
As expected, all methods exhibit some performance degradation under the condition of low signal-to-noise ratio. This is because the estimated number of targets given by the SMC-PHD filter becomes increasingly unreliable with an increase of detection uncertainty or the amount of clutters, which in turn has an influence on the state estimation. In addition, when the clutter rate increases, the probability that some clutter measurements appear at the regions of true targets or the birth intensities will increase correspondingly. It is possible that the contributions of these measurements are significant as those of true measurements. Therefore, the measurement-oriented methods (grouping method, MEAP method and Algorithm 1) are prone to be influenced by such clutter measurements. Obviously, the performance degradation of the grouping method is more apparent. Both the results in
Figure 5 and
Figure 6 demonstrate that the proposed methods yield the best performance in terms of estimation accuracy and robustness. Meanwhile, the improvement of Algorithm 2 seems limited in the case of lower detection uncertainty due to the following reasons: the SMC-PHD filter will lose tracking of targets frequently in such cases, while the estimation of undetected targets is performed under the premise that the corresponding particles exist in the state space. Thus, the improvement tends to be weakened when considering the global effect via evaluating the time-average of the OSPA metric.