Abstract
By integrating the cardinality balanced multitarget multiBernoulli (CBMeMBer) filter with the interacting multiple models (IMM) algorithm, an MMCBMeMBer 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 multitarget models and the Gaussian mixture (GM) method is used to implement the filter for linearGaussian multitarget models. Then, the extended Kalman (EK) and unscented Kalman filtering approximations for the GMMMCBMeMBer filter to accommodate mildly nonlinear models are described briefly. Simulation results are presented to show the effectiveness of the proposed filter.
1. Introduction
Recently, the randomfiniteset(RFS) based multitarget tracking approaches [1] have attracted extensive attention. Although theoretically solid, the RFSbased approaches usually are involved with intractable computations. By introducing the finiteset statistics (FISST) [2], Mahler developed the probability hypothesis density (PHD) [3] and cardinalized PHD (CPHD) [4] 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 [5], Sidenbladh [6], and Vo et al. [7]. Vo et al. and Zhang et al. [8–10] devised the Gaussian mixture (GM) implementations for the PHD and CPHD filters under the linearGaussian assumption on target dynamics, birth process, and sensor model. The PHDbased approaches have been successfully used for many realworld problems [11–13]. However, the SMCPHD and SMCCPHD approaches require clustering to extract state estimates from the particle population, which is expensive and unreliable [14, 15].
In 2007, Mahler proposed the multitarget multiBernoulli (MeMBer) [2] recursion, which is an approximation to the full multitarget Bayes recursion using multiBernoulli RFSs under low clutter density scenarios. In 2009, Vo et al. showed that the MeMBer filter overestimates the number of targets and proposed a cardinalitybalanced MeMBer (CBMeMBer) filter [16] to reduce the cardinality bias. Then, the SMC and GM implementations for the MeMBer and CBMeMBer filters were, respectively, proposed for generic and linearGaussian 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 multiBernoulli posterior density. Therefore, the key advantage of the SMCCBMeMBer filter over the SMCPHD and SMCCPHD filters is that the multiBernoulli 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 [17].
The original CBMeMBer filter does not consider the target maneuvers. Maneuvering targets might switch between different models of operation, so tracking using a singlemodel 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 [18] have been proven to be very effective and have better performance than the singlemodel 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 modelconditioned estimates; (2) modelconditioned basestate estimation; (3) deriving the overall state estimate by combining the estimates from each modelconditioned basestate filters.
By integrating the CBMeMBer filter with the IMM algorithm, an MMCBMeMBer 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 linearGaussian multitarget models. Then, the extended Kalman (EK) [19] and unscented Kalman (UK) [20] filtering approximations for the GMMMCBMeMBer filter to accommodate mildly nonlinear models are described briefly. Nonlinear and linearGaussian examples of multiple maneuvering targets tracking are, respectively, presented for comparing the performance of the MMCBMeMBer filter with that of the singlemodel CBMeMBer filters, MMPHD filter [21–24], and MMCPHD filter [25]. 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 singlemodel CBMeMBer filters is rather poor; (2) under relatively low clutter density, the SMCMMCBMeMBer filter outperforms the SMCMMPHD and SMCMMCPHD filters; (3) the performance of the GMMMCBMeMBer filter is similar to that of the GMMMPHD filter and hence is inferior to that of GMMMCPHD filter.
The rest of the paper is organized as follows. Section 2 describes the problem of multiple maneuvering targets tracking. In Section 3, the MMCBMeMBer recursion is given. The generic SMC implementation of the MMCBMeMBer filter is described in Section 4. The analytic GM implementation of the MMCBMeMBer filter for linearGaussian 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 targetgenerated measurements are indistinguishable from the clutter. At time , let denote the measurement vector received by a sensor. The singlemeasurement singletarget 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. MMCBMeMBer 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 multiBernoulli RFS is a union of a fixed number of independent Bernoulli RFSs , , that is, . is thus completely described by the multiBernoulli parameter set with the mean cardinality and the probability density [2] 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 multiBernoulli 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 MMCBMeMBer filter, which implicitly requires a finite number of singlemodel CBMeMBer filters operate in parallel, is derived by introducing the mixing and combination strategies in the IMM approaches [18]. As the multiplemodel approaches, the MMCBMeMBer filter does not need a maneuver detection decision and undergoes a soft switching between the models. One cycle of the recursive MMCBMeMBer algorithm can be described as follows.
(1) The Mixing and Prediction Stage. If at time , the posterior density is a multiBernoulli of the form , then the mixed multiBernoulli 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 modeldependent densities. Finally, the mixed and predicted density is also a multiBernoulli and is given by where are the parameters of the multiBernoulli 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 multiBernoulli of the form , then the posterior density can be approximated by a multiBernoulli as follows: where
(3) The Multitarget State Estimation. For the multiBernoulli 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 MMCBMeMBer filter [8] is summarized in Algorithm 1.

4. SMCMMCBMeMBer Filter
In this section, a generic SMC implementation of the proposed MMCBMeMBer filter is presented for accommodating nonlinear dynamic and measurement models. In this implementation, the samples or particles, which are used to represent the multiBernoulli density of multiple maneuvering targets, consists of the kinematical state and model information with associated weights. One cycle of the recursive SMCMMCBMeMBer algorithm can be described as follows.
(1) The SMC Mixing and Prediction Stage. Suppose that at time the multiBernoulli 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 multiBernoulli 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 multiBernoulli density is given and each , , is composed of a set of weighted samples ,
Then, the multiBernoulli approximation of the updated density can be computed as follows:
where
(3) The Resampling and Pruning Stage. It is the same as the resampling and pruning stage of the SMCCBMeMBer filter [16].
(4) The SMC Multitarget State Estimation. Given the SMC multiBernoulli 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 [26] can be introduced for increasing the particle variety after the resample step without affecting the validity of the SMC approximation.
5. GMMMCBMeMBer Filter and Its EK and UK Extensions
An analytic solution to the MMCBMeMBer recursion for linearGaussian multiple maneuvering targets models is presented in this section. The resulting filter propagates the GM multiBernoulli density against time. Some certain assumptions about the linearGaussian 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 multiBernoulli 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 MMCBMeMBer recursion, namely, the GMMMCBMeMBer filter, can be derived by applying the following two standard results for Gaussian functions: where
One cycle of the recursive GMMMCBMeMBer algorithm can be described as follows.
(1) The GM Mixing and Prediction Stage. Suppose that at time the multiBernoulli posterior density is given and each , , is composed of GM of the form Then, the mixed and predicted multiBernoulli density can be computed as follows: where
(2) The GM Update Stage. Suppose that at time the mixed and predicted multiBernoulli density is given and each , , is composed of GM of the form Then, the multiBernoulli approximation of the updated density can be computed as follows:
where
(3) The Pruning and Merging Stage. It is the same as the pruning and merging stage of the GMCBMeMBer filter [16].
(4) The GM Multitarget State Estimation. Given the GM multiBernoulli 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 GMMMCBMeMBer 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 modeldependent nonlinear functions, and and are modeldependent process and observation noise vectors of known statistics.
For the EKGMMMCBMeMBer 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 GMMMCBMeMBer filter with the corresponding local linearization of the nonlinear dynamical and observation models
Note that the unscented Kalman version for the GMMMCBMeMBer filter can be derived by approximating the mean and covariance of individual Gaussian components with a set of sigma points and the unscented transform [20]. Because of the space limitation, the details of the UKGMMMCBMeMBer filter are not presented here.
6. Simulations
6.1. Nonlinear Example Using SMC Implementations
In this nonlinear example, we evaluate the performance of the proposed MMCBMeMBer filter by benchmarking it against the singlemodel CBMeMBer filters, the MMPHD filter, and the MMCPHD filter using the SMC implementations.
Consider a twodimensional 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 [27]. 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 modeldependent dynamics for the individual maneuvering target is given by the linearGaussian 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