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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 727430, 16 pages
Multiple-Model Cardinality Balanced Multitarget Multi-Bernoulli Filter for Tracking Maneuvering Targets
Ministry of Education Key Laboratory for Intelligent Networks and Network Security (MOE KLINNS), School of Electronics and Information Engineering, Xi'an Jiaotong University, Xi'an 710049, China
Received 10 July 2013; Accepted 15 September 2013
Academic Editor: Xianxia Zhang
Copyright © 2013 Xianghui Yuan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
By integrating the cardinality balanced multitarget multi-Bernoulli (CBMeMBer) filter with the interacting multiple models (IMM) algorithm, an MM-CBMeMBer filter is proposed in this paper for tracking multiple maneuvering targets in clutter. The sequential Monte Carlo (SMC) method is used to implement the filter for generic multi-target models and the Gaussian mixture (GM) method is used to implement the filter for linear-Gaussian multi-target models. Then, the extended Kalman (EK) and unscented Kalman filtering approximations for the GM-MM-CBMeMBer filter to accommodate mildly nonlinear models are described briefly. Simulation results are presented to show the effectiveness of the proposed filter.
Recently, the random-finite-set-(RFS-) based multitarget tracking approaches  have attracted extensive attention. Although theoretically solid, the RFS-based approaches usually are involved with intractable computations. By introducing the finite-set statistics (FISST) , Mahler developed the probability hypothesis density (PHD)  and cardinalized PHD (CPHD)  filters, which have been shown to be a computationally tractable alternative to full multitarget Bayes filters in the RFS framework. The sequential Monte Carlo (SMC) implementations for the PHD and CPHD filters were devised by Zajic and Mahler , Sidenbladh , and Vo et al. . Vo et al. and Zhang et al. [8–10] devised the Gaussian mixture (GM) implementations for the PHD and CPHD filters under the linear-Gaussian assumption on target dynamics, birth process, and sensor model. The PHD-based approaches have been successfully used for many real-world problems [11–13]. However, the SMC-PHD and SMC-CPHD approaches require clustering to extract state estimates from the particle population, which is expensive and unreliable [14, 15].
In 2007, Mahler proposed the multitarget multi-Bernoulli (MeMBer)  recursion, which is an approximation to the full multitarget Bayes recursion using multi-Bernoulli RFSs under low clutter density scenarios. In 2009, Vo et al. showed that the MeMBer filter overestimates the number of targets and proposed a cardinality-balanced MeMBer (CBMeMBer) filter  to reduce the cardinality bias. Then, the SMC and GM implementations for the MeMBer and CBMeMBer filters were, respectively, proposed for generic and linear-Gaussian dynamic and measurement models. The MeMBer and CBMeMBer recursions propagate not the moments and cardinality distributions which are propagated by the PHD and CPHD filters but rather the approximate multitarget multi-Bernoulli posterior density. Therefore, the key advantage of the SMC-CBMeMBer filter over the SMC-PHD and SMC-CPHD filters is that the multi-Bernoulli representation of the posterior density allows reliable and inexpensive extraction of state estimates. The CBMeMBer filter has been applied for tracking multiple targets according to their audio and visual information .
The original CBMeMBer filter does not consider the target maneuvers. Maneuvering targets might switch between different models of operation, so tracking using a single-model CBMeMBer filter might fail since the filter does not match the actual system dynamics. It is well known that the interacting multiple models (IMM) approaches  have been proven to be very effective and have better performance than the single-model filters in tracking a single maneuvering target without clutter. In the IMM approaches, a bank of filters, each matched with a different target motion model, operate in parallel. In general, there are three key steps in the IMM estimators: (1) mixing the model-conditioned estimates; (2) model-conditioned base-state estimation; (3) deriving the overall state estimate by combining the estimates from each model-conditioned base-state filters.
By integrating the CBMeMBer filter with the IMM algorithm, an MM-CBMeMBer filter is proposed to address the problem of tracking multiple maneuvering targets in clutter, which is much more difficult than the problem of tracking a single maneuvering target without clutter since the association between the measurements and the targets is unknown. The SMC method is used to implement the filter for generic multitarget models while the GM method is used to implement the filter for linear-Gaussian multitarget models. Then, the extended Kalman (EK)  and unscented Kalman (UK)  filtering approximations for the GM-MM-CBMeMBer filter to accommodate mildly nonlinear models are described briefly. Nonlinear and linear-Gaussian examples of multiple maneuvering targets tracking are, respectively, presented for comparing the performance of the MM-CBMeMBer filter with that of the single-model CBMeMBer filters, MM-PHD filter [21–24], and MM-CPHD filter . The simulation results show that (1) the proposed filter can estimate the number and states of multiple maneuvering targets effectively, whereas the performance of the single-model CBMeMBer filters is rather poor; (2) under relatively low clutter density, the SMC-MM-CBMeMBer filter outperforms the SMC-MM-PHD and SMC-MM-CPHD filters; (3) the performance of the GM-MM-CBMeMBer filter is similar to that of the GM-MM-PHD filter and hence is inferior to that of GM-MM-CPHD filter.
The rest of the paper is organized as follows. Section 2 describes the problem of multiple maneuvering targets tracking. In Section 3, the MM-CBMeMBer recursion is given. The generic SMC implementation of the MM-CBMeMBer filter is described in Section 4. The analytic GM implementation of the MM-CBMeMBer filter for linear-Gaussian multitarget models and its EK and UK extensions for nonlinear multitarget models are, respectively, given in Section 5. Numerical studies are shown in Section 6. The conclusions and the future work are given in Section 7.
2. Problem Statement for Multiple Maneuvering Targets Tracking
The multiple maneuvering targets appear and disappear randomly against time over an observation region. At time , let denote the kinematical state of a target and the label of the model in effect, where is the discrete set of all model labels. The models follow a discrete Markov chain with transition probability . Let denote the augmented state vector, whose transition is governed by the density where is the kinematical state transition density conditioned on model .
The measurement originates either from target or from random clutter (false alarm). Moreover, the target-generated measurements are indistinguishable from the clutter. At time , let denote the measurement vector received by a sensor. The single-measurement single-target likelihood is described by the density conditioned on model
At time , let denote the number of the existing targets and the number of the measurements. Then, multiple augmented states and unlabelled sensor measurements can be represented as finite sets and , respectively. In addition, let denote a sequence of the measurement sets available up to and including time .
3. MM-CBMeMBer Filter
A Bernoulli RFS has probability of being empty and probability () of being a singleton whose only element is distributed according to a probability density . The probability density of is
A multi-Bernoulli RFS is a union of a fixed number of independent Bernoulli RFSs , , that is, . is thus completely described by the multi-Bernoulli parameter set with the mean cardinality and the probability density  where denotes the cardinality of a set.
Throughout this paper, we abbreviate a probability density of the form (4) by .
Let denote the probability that the maneuvering target with augmented state survives at time ; let denote the probability that the maneuvering target with augmented state generates an observation at time . RFS modeling the multiple maneuvering targets state and the sensor measurement are, respectively, given by the union where denotes the multi-Bernoulli RFS of spontaneous births; the Bernoulli RFS with and is used to model the dynamic behavior of ; the Bernoulli RFS with and is used to model the observation behavior of ; the clutter is modeled as a Poisson RFS with the intensity , where and are, respectively, the average clutter number and the probability density of clutter spatial distribution at time .
Based on the above RFS models of the multiple maneuvering targets and the method of Mahler’s FISST, the MM-CBMeMBer filter, which implicitly requires a finite number of single-model CBMeMBer filters operate in parallel, is derived by introducing the mixing and combination strategies in the IMM approaches . As the multiple-model approaches, the MM-CBMeMBer filter does not need a maneuver detection decision and undergoes a soft switching between the models. One cycle of the recursive MM-CBMeMBer algorithm can be described as follows.
(1) The Mixing and Prediction Stage. If at time , the posterior density is a multi-Bernoulli of the form , then the mixed multi-Bernoulli density is where
Since the models switching is only decided by the model transition probability and is independent of the target kinematical state: is a combination of the previous model-dependent densities. Finally, the mixed and predicted density is also a multi-Bernoulli and is given by where are the parameters of the multi-Bernoulli RFS of births at time : where defines the integral inner product, that is,
(2) The Update Stage. If at time , the mixed and predicted density is a multi-Bernoulli of the form , then the posterior density can be approximated by a multi-Bernoulli as follows: where
(3) The Multitarget State Estimation. For the multi-Bernoulli representation , the extraction of multitarget number and state estimates are straightforward since the probability indicates how likely the th hypothesized track is a true track, and the posterior density describes the distribution of the estimated augmented state of the track. The state estimation procedure for the MM-CBMeMBer filter  is summarized in Algorithm 1.
4. SMC-MM-CBMeMBer Filter
In this section, a generic SMC implementation of the proposed MM-CBMeMBer filter is presented for accommodating nonlinear dynamic and measurement models. In this implementation, the samples or particles, which are used to represent the multi-Bernoulli density of multiple maneuvering targets, consists of the kinematical state and model information with associated weights. One cycle of the recursive SMC-MM-CBMeMBer algorithm can be described as follows.
(1) The SMC Mixing and Prediction Stage. Suppose that at time the multi-Bernoulli posterior density is given and each , , is composed of a set of weighted samples , where is Dirac delta function centered at . Then, the mixed and predicted multi-Bernoulli density can be computed as follows: where the particles , corresponding to the surviving maneuvering targets can be derived by sampling from the proposal densities and with the associated weights and the particles , corresponding to the new born maneuvering targets can be derived by sampling from the proposal densities and with the associated weights
(2) The SMC Update Stage. Suppose that at time the mixed and predicted multi-Bernoulli density is given and each , , is composed of a set of weighted samples ,
Then, the multi-Bernoulli approximation of the updated density can be computed as follows:
(3) The Resampling and Pruning Stage. It is the same as the resampling and pruning stage of the SMC-CBMeMBer filter .
(4) The SMC Multitarget State Estimation. Given the SMC multi-Bernoulli posterior density from the method described in Algorithm 1, the SMC multitarget state estimation can be easily obtained as
Note that the MCMC move step  can be introduced for increasing the particle variety after the resample step without affecting the validity of the SMC approximation.
5. GM-MM-CBMeMBer Filter and Its EK and UK Extensions
An analytic solution to the MM-CBMeMBer recursion for linear-Gaussian multiple maneuvering targets models is presented in this section. The resulting filter propagates the GM multi-Bernoulli density against time. Some certain assumptions about the linear-Gaussian multiple maneuvering targets models are firstly summarized below.
(A) The dynamic and measurement models for the augmented state of each maneuvering target have the form where denotes the density of Gaussian distribution with the mean and covariance ; , , and are, respectively, the kinematical state transition, process noise covariance, and process noise coefficient matrixes conditioned on model ; , , and are, respectively, the observation, observation noise covariance, and observation noise coefficient matrixes conditioned on model .
(B) The probabilities of maneuvering target survival and maneuvering target detection are independent of the kinematical state:
(C) The birth model for the maneuvering targets is a multi-Bernoulli with parameter set , where , , are GM of the form where is the distribution of model births and is the distribution of the birth kinematical state given model is GM of the form with the parameter set .
According to the above Assumptions A, B, and C, a closed form solution to the MM-CBMeMBer recursion, namely, the GM-MM-CBMeMBer filter, can be derived by applying the following two standard results for Gaussian functions: where
One cycle of the recursive GM-MM-CBMeMBer algorithm can be described as follows.
(1) The GM Mixing and Prediction Stage. Suppose that at time the multi-Bernoulli posterior density is given and each , , is composed of GM of the form Then, the mixed and predicted multi-Bernoulli density can be computed as follows: where
(2) The GM Update Stage. Suppose that at time the mixed and predicted multi-Bernoulli density is given and each , , is composed of GM of the form Then, the multi-Bernoulli approximation of the updated density can be computed as follows:
(3) The Pruning and Merging Stage. It is the same as the pruning and merging stage of the GM-CBMeMBer filter .
(4) The GM Multitarget State Estimation. Given the GM multi-Bernoulli posterior density from the method described in Algorithm 1, the GM multitarget state estimation can be easily obtained as
Now turn to considering the extension of the GM-MM-CBMeMBer filter to nonlinear dynamical and observation models using the EK filtering approximation. Assumptions B and C are still required, but the dynamic and observation processes can be relaxed to the nonlinear models where and are known model-dependent nonlinear functions, and and are model-dependent process and observation noise vectors of known statistics.
For the EK-GM-MM-CBMeMBer filter, the closed form expressions for the mixing, prediction, and update of individual Gaussian components are approximated by replacing , , , in the corresponding recursive equations (30)–(35) of the GM-MM-CBMeMBer filter with the corresponding local linearization of the nonlinear dynamical and observation models
Note that the unscented Kalman version for the GM-MM-CBMeMBer filter can be derived by approximating the mean and covariance of individual Gaussian components with a set of sigma points and the unscented transform . Because of the space limitation, the details of the UK-GM-MM-CBMeMBer filter are not presented here.
6.1. Nonlinear Example Using SMC Implementations
In this nonlinear example, we evaluate the performance of the proposed MM-CBMeMBer filter by benchmarking it against the single-model CBMeMBer filters, the MM-PHD filter, and the MM-CPHD filter using the SMC implementations.
Consider a two-dimensional scenario with an unknown and time varying number of the maneuvering targets observed over the region () for a period of time steps. The sampling interval is (). Each of the targets may move at a nearly constant velocity or execute a coordinated turn in the surveillance period. Therefore, the model set designed for this example can be composed of a constant velocity (CV) model and a coordinated turn (CT) model with varying turn rate . The target kinematical state is , where and , respectively, represent the position and the velocity in and coordinates and represents the turn rate. For the turn rate , let the anticlockwise direction be positive and the clockwise direction be negative.
The model-dependent dynamics for the individual maneuvering target is given by the linear-Gaussian model
Let denote the CV model and the CT model; then with where is the level of the power spectral density of the process noise for model . In this example, they are given by (), (), ().
The Markovian model transition probability matrix is taken as
At time , the range and bearing measurements of the targets are generated by a sensor located at . The measurement noise is independent and identically distributed (IID) zero-mean Gaussian white noise with covariance matrix , where denotes the diagonal matrix, and and are, respectively, standard deviations (STDs) of the range and bearing measurements. In this example, they are taken as () and (). The single-measurement single-target likelihood density is
The detection probability and the survival probability are, respectively, taken as and in this example.
The clutter is modeled as a Poisson RFS with the intensity . In this example, we take and , where denotes the density of the uniform distribution over the observation region.
Figure 1 shows the true trajectories for the maneuvering targets and sensor location.
In Figure 1, “” denotes the locations at which targets are born and “” denotes the locations at which targets die. Target 1 is born at 1 s and dies at 30 s. It first moves at a nearly constant velocity from the first second to the 15th second and then executes a coordinated turn in the anticlockwise direction from the 16th second to the 30th second. Target 2 is born at 1 s and dies at 35 s. It first executes a coordinated turn in the anticlockwise direction from the first second to the 20th second and then moves at a nearly constant velocity from the 21st second to the 35th second. Target 3 is born at 10 s and dies at 42 s. It first executes a coordinated turn in the anticlockwise direction from the 10th second to the 30th second and then moves at a nearly constant velocity from the 31st second to the 42nd second. Target 4 is born at 20 s and dies at 50 s. It first moves at a nearly constant velocity from the 20th second to the 30th second and then executes a coordinated turn in the clockwise direction from the 31st second to the 50th second. The motions of the targets are summarized in Table 1.
The birth process is a multi-Bernoulli RFS with density , where , , with and the distribution of the model births
For the purpose of comparison, we estimate the number and states of the maneuvering targets using the proposed SMC-MM-CBMeMBer filter, the CV model SMC-CBMeMBer filter, the CT model SMC-CBMeMBer filter, the SMC-MM-PHD filter, and the SMC-MM-CPHD filter, respectively. At each time step in the SMC implementations of the CBMeMBer-based filters, a maximum of and minimum of particles per hypothesized track are imposed, and pruning of hypothesized tracks is performed with a threshold of . At each time step in the SMC implementations of the PHD-based filters, 1000 particles are used to represent one target and -means method  is used to cluster the resampled particles to extract the multitarget states. The proposal densities , , , and in (16) and (18) are, respectively, taken as , , and . We now conduct 500 Monte Carlo (MC) simulation experiments on the same clutter intensity and target trajectories, but with independently generated clutter and target-generated measurements in each trial.
The MC averages of the mean and STD of the cardinality distribution for the five methods at each time step are shown along with the true target number in Figure 2, respectively.
Figures 2(a)–2(e) demonstrate that the target number estimates from the SMC-MM-PHD, SMC-MM-CPHD, and SMC-MM-CBMeMBer filters converge to the ground truth, whereas the CV model SMC-CBMeMBer and CT model SMC-CBMeMBer filters produce significant bias in estimating the target number. This is because the SMC-MM-PHD, SMC-MM-CPHD, and SMC-MM-CBMeMBer filters can effectively capture the model switching property of the maneuvering targets, so their performance is significantly better than that of the two single-model SMC-CBMeMBer filters, which show poor adaptation to target maneuvers and yield larger estimation errors.
Moreover, as plotted in Figures 2(c)–2(e), the STD of the cardinality distribution from the SMC-MM-CBMeMBer filter is lower than that of the SMC-MM-PHD filter, but larger than that of SMC-MM-CPHD filter. In addition, the STDs of the cardinality distributions from the three MM-based filters increase in different degrees at the instances when the maneuver occurs (i.e., 16 (), 21 (), and 31 ()). The STD plots of the SMC-MM-PHD and SMC-MM-CPHD filters seem to fluctuate more obviously than the SMC-MM-CBMeMBer filter. This phenomenon indicates that the performance of the SMC-MM-CBMeMBer filter may be more stable and robust at the maneuver instances than that of the SMC-MM-PHD and SMC-MM-CPHD filters.
The optimal subpattern assignment (OSPA) metric , which can jointly capture differences in cardinality and individual elements between two finite sets, is used to evaluate the performance of the five methods. Given the actual and estimated multitarget state sets and , the OSPA metric of order with cut-off between the two sets is defined by if and if . denotes the set of permutations on . denotes the 2-norm. In this example, we take ().
The MC averages of the OSPA metric for the target position estimates, derived by the five methods, are shown in Figure 3.
The OSPA metric is composed of two components each separately accounting for “localization” and “cardinality” errors. This results in high peaks in OSPA metric at the instances where the estimated number is incorrect. Figure 3 shows that (1) both the single-model SMC-CBMeMBer filters perform significantly worse than the other MM-based filters because of the large cardinality errors produced by the two filters as seen in Figures 2(a) and 2(b); (2) although the SMC-MM-CPHD filter can estimate the target number most accurately, the OSPA metric of the SMC-MM-CBMeMBer filter is smaller than that of the SMC-MM-CPHD filter, which is in turn smaller than that of the SMC-MM-PHD filter. This phenomenon indicates that the SMC-MM-CBMeMBer filter outperforms the SMC-MM-CPHD (and hence SMC-MM-PHD) filter in jointly estimating the multitarget number and states. A reason for this is that the additional errors could be introduced in the clustering processes of the SMC-MM-PHD and SMC-MM-CPHD filters to extract state estimates from the particle population; (3) the OSPA plots of the three MM-based filters in Figure 3 fluctuate against time due to the varying target number, the target maneuvers, and clutter. However, increase of the OSPA from the SMC-MM-CBMeMBer filter seems to be smallest at the maneuver instances (i.e., 16 (), 21 (), and 31 ()) among the three MM-based methods. This phenomenon also indicates that the performance of the SMC-MM-CBMeMBer filter may be more stable and robust at the maneuver instances than that of the SMC-MM-PHD and SMC-MM-CPHD filters.
For comparing the overall performance of the three MM-based filters, the 500 MC trial averages of the OSPA distance (time-averaged over the duration of the scenario) for the three MM-based filters are shown in Table 2 against the clutter rate from to . The result of time-averaging can be viewed as a broad indication of filter performance, although the average is likely to be scenario dependent.
Table 2 shows that the OSPA distances of the three MM-based filters increase with higher . It reflects that the performance of the three MM-based algorithms degrades by different degrees as the increases. Among the three MM-based algorithms, the SMC-MM-PHD filter always works the worst. The SMC-MM-CBMeMBer filter outperforms the SMC-MM-CPHD filter when is relatively lower (). However, as the increases, the OSPA distance of SMC-MM-CBMeMBer filter increases more rapidly than that of the SMC-MM-CPHD filter. Therefore, as continues to increase until it reaches , the OSPA distance of SMC-MM-CBMeMBer filter is very close to that of the SMC-MM-CPHD filter. When is relatively higher (i.e., ), the SMC-MM-CPHD filter outperforms the SMC-MM-CBMeMBer filter. A possible reason for this is that, compared with the SMC-MM-CBMeMBer filter, the advantage of the target number estimate for the SMC-MM-CPHD filter is more obvious as the increases and it finally leads that the OSPA distance of the SMC-MM-CPHD filter is smaller than that of the SMC-MM-CBMeMBer filter when is relatively higher (i.e., ).
6.2. Linear-Gaussian Example Using GM Implementations
In this linear-Gaussian example, we evaluate the performance of the proposed MM-CBMeMBer filter by benchmarking it against the single-model CBMeMBer filters, the MM-PHD filter, and the MM-CPHD filter using the GM implementations.
The simulation scenario and true trajectories for the maneuvering targets are the same as those of Example 1. The target kinematical state now turns into . The model set for this example is designed as follows. Model is a CV model with linear-Gaussian dynamics given by ; models are, respectively, CT models with turn rates of () with linear-Gaussian dynamics given by . In this example, and are given by (),