Sequential measurementdriven multitarget Bayesian filter
 Zongxiang Liu^{1}Email author,
 Lijuan Li^{1},
 Weixin Xie^{1} and
 Liangqun Li^{1}
https://doi.org/10.1186/s1363401502288
© Liu et al. 2015
Received: 16 January 2015
Accepted: 23 April 2015
Published: 15 May 2015
Abstract
Bayesian filter is an efficient approach for multitarget tracking in the presence of clutter. Recently, considerable attention has been focused on probability hypothesis density (PHD) filter, which is an intensity approximation of the multitarget Bayesian filter. However, PHD filter is inapplicable to cases in which target detection probability is low. The use of this filter may result in a delay in data processing because it handles received measurements periodically, once every sampling period. To track multiple targets in the case of low detection probability and to handle received measurements in real time, we propose a sequential measurementdriven Bayesian filter. The proposed filter jointly propagates the marginal distributions and existence probabilities of each target in the filter recursion. We also present an implementation of the proposed filter for linear Gaussian models. Simulation results demonstrate that the proposed filter can more accurately track multiple targets than the Gaussian mixture PHD filter or cardinalized PHD filter.
Keywords
Multitarget tracking Bayesian filter Probability hypothesis density filter Marginal distribution Existence probability1 Introduction
Multitarget tracking aims to detect individual targets in the surveillance region of interest and estimate their states according to a sequence of noisy and cluttered measurements collected by sensors. The most efficient technique for multitarget tracking is the multitarget Bayesian filter, which propagates joint posterior distribution of the multitarget state [1, 2]. However, such propagation is computationally intensive because of the high dimensionality of the multitarget state space [2, 3]. With the use of the Bayesian framework to propagate the posterior intensity of multiple targets recursively, the probability hypothesis density (PHD) filter provides a numerically tractable solution to this problem [2, 3]. Two numerical solutions, namely sequential Monte Carlo (SMC) [4–9] and Gaussian mixtures (GM) [10–17], have been developed for the PHD filter. Extensions of the PHD filter have also been proposed to improve its performance. PHD filters with observationdriven birth intensity were independently proposed in [16, 18, 19] to eliminate the need for exact knowledge of birth intensity. Methods for maintaining track continuity were proposed in [4, 20] for the SMCPHD filter and in [21] for the GMPHD filter. To improve the accuracy and stability of the target number estimate, the cardinalized PHD (CPHD) filter, which jointly propagates moment and cardinality, was proposed in [22]. Methods for estimating an unknown clutter rate, which is an important parameter of the PHD and CPHD filters, were proposed in [23] and [24]. In [12], the GMPHD filter was extended to linear jump Markov multitarget models for use in tracking maneuvering targets.
Although the PHD filter has several advantages, it becomes inefficient in cases with low target detection probability. This inefficiency occurs because the PHD filter has a weak memory and is easily influenced by new incoming measurements [2, 17, 22]. Owing to its weak memory, the PHD filter fails to provide state estimates of existing targets if these targets are missing from new incoming measurements [2]. Moreover, the PHD filter may result in data processing delay. This delay occurs because the PHD filter handles new incoming measurements periodically, once every sampling period. In this manner, new measurements have to be gathered for a sampling period before being processed. Therefore, a significant delay arises from a long sampling period. The CPHD filter has a better memory than the PHD filter but has a slower response in detecting target appearance and disappearance and is less influenced by new incoming measurements. Despite its strong memory, the CPHD filter may also be inefficient in cases with low detection probability because it has more difficulty than the PHD filter in giving state estimates of new targets because of its slow response to new targets. In addition, similar to the PHD filter, the CPHD filter may also result in data processing delay because it also handles new incoming measurements periodically.
To resolve the multitarget tracking problem efficiently in the case of low detection probability and to reduce data processing delay, we propose a sequential measurementdriven Bayesian filter. This filter propagates the marginal distributions and existence probabilities of each target in the filter recursion and uses received measurements to generate new marginal distributions and update existing marginal distributions. In this filter, we use the prediction marginal distribution and existence probability of a target as its update marginal distribution and existence probability, respectively, and propagate them in the filter recursion if this target is missing from the incoming measurements. Thus, the proposed filter has a sufficient memory to missing targets, which enables this filter applicable to tracking multiple targets in the case of low detection probability. Moreover, this filter reduces the data processing delay that exists in PHD and CPHD filters because new incoming measurements can be processed whenever they become available. We also propose a closedform solution and implementation of the proposed filter for linear Gaussian models. In terms of optimal subpattern assignment (OSPA) distance [25], we compare the proposed filter with the PHD and CPHD filters through simulation. These three filters are capable of tracking multiple targets in the presence of clutter as well as target appearance and disappearance. Among the three filters, the proposed filter is the best at tracking multiple targets in cases of low detection probability.
The main contributions of this paper are twofold. First, we propose a new Bayesian filter for sequentially handling new measurements. The proposed filter handles received measurements in real time and is applicable to tracking multiple targets in the case of low detection probability. Second, we present a novel implementation of the proposed filter for linear Gaussian models.
The remainder of this paper is organized as follows: Section 2 briefly introduces the multitarget Bayesian filter. Section 3 proposes the sequential measurementdriven multitarget Bayesian filter to propagate marginal distribution. Section 4 discusses the implementation of the proposed filter in linear Gaussian models. Section 5 evaluates the performance of the proposed filter. Section 6 states the drawn conclusions.
2 Multitarget Bayesian filter
We first provide a brief description of the multitarget Bayesian filter [1]. In a multitarget Bayesian filter, the distribution of interest is the joint posterior f(x _{ k }y _{1 : k }), which is also known as the filtering distribution, where k denotes the time step, x _{ k } = (x _{1,k } ⋯ x _{ N ,k }) is the multitarget state at time step k, N is the target number, and y _{1 : k } = (y _{1} ⋯ y _{ k }) represents all observations from time step 1 to time step k. The filtering distribution of a multitarget Bayesian filter can be computed by using a twostep recursion.
Equation 2 clearly shows that the new filtering distribution is obtained by directly applying the Bayesian rule. The generally intractable multitarget Bayesian filter involves integrals of high dimensions. This intractability is usually resolved by applying fixed grid approximation, SMC approximation, and PHD approximation [1, 2].
3 Sequential measurementdriven multitarget Bayesian filter

A1. Targets evolve and generate observations independent of one another.

A2. Clutter is independent of targetoriginated measurements.

A3. The survival and detection probabilities of each target are state independent.
To derive the SMB filter conveniently, we also assume that the number of targets at time step k − 1 is N _{ k − 1}; the states of individual targets at time step k − 1 are x _{ i,k − 1}, i = 1 ⋯ N _{ k − 1}; the marginal distributions and existence probabilities of individual targets at time step k − 1 are f _{ i }(x _{ i,k − 1}y _{1 : k − 1}), i = 1 ⋯ N _{ k − 1} and p _{i,k − 1}, i = 1 ⋯ N _{ k − 1}, respectively; and y _{1 : k − 1} = (y _{1} ⋯ y _{ k − 1}) represents all observations up to time step k − 1. The objective is to determine the marginal distributions for each target and their existing probabilities at time step k.
From Equations 6, 7, 8, 9, 10, 11, and 12, it is clear that f _{ i }(x _{ i,k }y _{1 : k }) = f _{ i }(x _{ i,k }y _{1 : k − 1}) and p _{ i,k } = p _{ i,kk − 1} if no measurement is originated from target i. This phenomenon indicates that this filter uses the prediction marginal distribution and existence probability of a target as its update marginal distribution and existence probability, respectively, if the target is missing from the incoming measurements and also indicates that this filter has a sufficient memory to missing targets.
Please note that individual targets are not tracked independently in the proposed filter. As shown in Equation 6, we use the marginal distributions f _{ e,j − 1}(x _{ e,k }), e = 1 ⋯ N _{ k − 1} of individual targets to obtain the update distribution \( {f}_{i,j}^u\left({\mathbf{x}}_{i,k}\right) \) of target i. Therefore, individual targets are dependent in this filter.
where N _{ k } = N _{ k − 1} + M; \( {f}_{\gamma}^j\left({\mathbf{x}}_{i,k}\right) \), j = 1 ⋯ M are the marginal distributions of the new targets; and \( {p}_{\gamma}^j \), j = 1 ⋯ M are the existence probabilities of the new targets.
Posterior distributions f _{ i }(x _{ i,k }y _{1 : k }), i = 1 ⋯ N _{ k } in Equation 13 are marginal distributions at time step k, and existence probabilities p _{ i,k }, i = 1 ⋯ N _{ k } in Equation 14 are existence probabilities of the marginal distributions at time step k.
4 Implementation of the SMB filter for linear Gaussian models

A4: Each target follows a linear Gaussian dynamic model, and the sensor has a linear Gaussian measurement model, i.e.,

A5: The new marginal distributions at time step k are Gaussian distribution and are generated from observations at time step k as:
Any measurement received at time step k may be originated from a new target, an existing target, or clutter. Instead of classifying a measurement as belonging to a new target, we use each measurement to generate a new distribution and assign a small existence probability to each new distribution in the filter recursion.

Prediction step: Given that the marginal distributions of individual targets at time step k−1 are:

Update step: In this step, we deal with the received observations y _{ k } = (y _{1,k } ⋯ y _{ M,k }) one after another by using the Bayesian rule to obtain individual update distributions and their corresponding existence probabilities.

Multitarget state extraction step: Given the extended update distributions f _{ i }(x _{ i,k }y _{1 : k }) = N(x _{ i,k }; m _{ i,k }, P _{ i,k }), i = 1 ⋯ N _{ k } and their existence probabilities p _{ i,k }, i = 1 ⋯ N _{ k }, we first eliminate the Gaussian distribution whose existence probability is smaller than a given threshold τ. After pruning, the remaining Gaussian distributions and their existence probabilities are used as inputs for the next filtering recursion. We then select the Gaussian distributions with existence probabilities p _{ i,k } > 0.5 as outputs of the filter. The mean of a selected Gaussian distribution is the state estimate of a target.
5 Simulation results
We select the PHD/CPHD filters with measurementdriven birth intensity [19] as two contesting comparison objects in the simulation. The selected filters adapt the target birth intensity at each processing step with the use of the received measurements, thereby eliminating the need for prior specification of birth intensities [19].
We consider an example in this section. This example shows the tracking performances of the PHD filter, the CPHD filter, and the proposed SMB filter for linear Gaussian models. In this example, the pruning procedure presented in [10] is performed at each time step of both GMPHD filter and GMCPHD filter. The state extraction in GMPHD filter involves selecting the means of the Gaussians that have weights greater than 0.5 as the state estimates [10], whereas the state extraction in GMCPHD filter involves estimating the number of targets and then extracting the corresponding number of Gaussians with the highest weights as state estimates [22]. To assess the tracking performances of these three filters, we use the OSPA distance [25] as the measure, where parameters are set as p = 2 and c = 50 m.
where σ _{ w } is the standard deviation of the observation noise.
To detect the targets in the surveillance space and to estimate their states from the measurement data, we set the corresponding parameters of these three filters to λ _{ c,k } = 5 × 10^{−6} m^{−2}, p _{ D,k } = 0.8, σ _{ v } = 1 ms^{−2}, σ _{ w } = 2 m, τ = 10^{−3}, and δ = 2. The weight \( {w}_{\gamma}^j \) of the new distribution in Gaussian mixture PHD/CPHD filters and the existence probability \( {p}_{\gamma}^j \) of the new distribution in the SMB filter are set to \( {w}_{\gamma}^j={p}_{\gamma}^j=0.05 \). The covariance of the new distribution is \( {\mathbf{P}}_{\gamma}^j={\left(\mathrm{diag}\left(\left[50\quad 25 \quad 50 \quad 25\right]\right)\right)}^2 \). In both PHD and CPHD filters, the survival probability p _{ S,k } is set to p _{ S,k } = 1.0.
Average performing time (s) of a Monte Carlo run for different detection probabilities
p_{D,k}  1.0  0.95  0.9  0.85  0.8  0.75  0.7  0.65 

SMB filter  23.21  22.44  22.02  21.73  21.36  20.88  20.35  19.75 
PHDM filter  1.84  2.46  3.09  4.09  4.29  4.98  6.25  6.50 
CPHDM filter  28.13  29.39  29.75  29.87  29.07  28.71  28.27  27.91 
Average performing time ( s ) of a Monte Carlo run for different clutter rates
λ _{ c,k }(m^{−2})  0.00000125  0.0000025  0.00000375  0.000005  0.00000625 

SMB filter  7.90  12.37  16.65  21.36  25.78 
PHDM filter  0.79  1.48  2.37  4.29  7.03 
CPHDM filter  14.48  19.89  24.16  29.07  33.22 
6 Conclusions
In this study, we propose a sequential measurementdriven multitarget Bayesian filter. This filter propagates marginal distributions and existence probabilities for each target. We also present an implementation of the proposed filter for linear Gaussian models. The proposed filter can process new observations as soon as these new observations become available, thereby reducing data processing delay. This filter can track multiple targets in the presence of clutter as well as target appearance and disappearance and is applicable to cases in which the detection probability of the target is low and the sampling period is long. Simulation results show that this filter tracks multiple targets better than the PHD and CPHD filters given a low detection probability of the target.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Nos. 61271107 and 61301074), Shenzhen Basic Research Project (No. JCYJ20140418095735618), and National Defense Preresearch Foundation (No. 9140C800501140C80340).
Authors’ Affiliations
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