PHD filter for multi-target tracking with glint noise
Introduction
Multi-target tracking has received great attention due to its wide applications in military and civil fields. As new targets may appear or disappear randomly in the surveillance region, tracking multi-target involves estimating an unknown and time-varying number of targets from a given set of measurements with uncertain origin. Many tracking algorithms have been proposed based on data association strategies such as the nearest neighbor (NN), the strongest neighbor (SN), the joint probabilistic data association (JPDA) and the multiple hypothesis tracking (MHT) [1]. Due to its combinatorial nature, a high computational load is often required to resolve the data association problem in multi-target tracking algorithms. Two well-known alternative formulations that avoid data association are symmetric measurement equations [2] and the random finite sets (RFS) approach [3].
In the RFS formulation, the target states and measurements are represented by two different random finite sets (RFSs), and as a consequence, the multi-target tracking problem can be addressed in a rigorous Bayesian estimation framework based on the finite set statistics (FISST) theory. By constructing the multi-target transition density and the multi-target likelihood function, the optimal multi-target Bayes filter can be derived. However, it is generally intractable due to the existence of multiple set integrals and the combinatorial nature of the multi-target densities. To alleviate this intractability, the probability hypothesis density (PHD) filter has been proposed as a first order moment approximation to the multi-target posterior density [4]. It is worth mentioning that the resulting PHD filter still requires solving multi-dimensional integrals and the integrals might be also intractable in many cases of interest. Two schemes have been proposed to implement the PHD filter explicitly including the sequential Monte Carlo (SMC) [5] and the Gaussian mixture (GM) [6]. A main drawback of the SMC-PHD filter is high computational cost since a large number of particles have to be sampled. To overcome this disadvantage, the GM-PHD filter was developed for linear target dynamic and measurement models with Gaussian distributions, in which the weights, means and covariance matrices are propagated analytically by the Kalman filter (KF). Moreover, the nonlinear KF counterparts can be directly employed to deal with nonlinear target dynamics and measurement models. The convergence properties of two implementations were analyzed in [5], [7]. Many extensions have been recently developed to address different tracking scenarios [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20].
For multi-target tracking problems, the Gaussian distribution has commonly been used for representing the measurement noise statistics due to its mathematical simplicity and effectiveness. However, it is known that not all the real-world data can be modeled well by Gaussian distribution. In particular for radar tracking systems, changes in the target aspect toward the radar may cause irregular electromagnetic wave reflections and this gives rise to outliers or glint noise. It was found that glint noise has a heavy-tailed probability density function and conventional filtering algorithms are known to show unsatisfactory performance in the presence of glint noise [21]. The statistical properties of the glint noise and its mathematical models have been studied extensively in [22]. Specially, the Student's t-distribution has been used to model the glint noise in [23] while the mixture of Gaussian distributions has been used in [24]. In [25], the glint noise was modeled by the mixture of a Gaussian distribution and a Laplacian distribution. As the Student's t-distribution has been shown to be much less sensitive than the Gaussian distribution to outliers [26], [27], [28], it has been used as an image prior [29], [30], [31]. A robust multi-sensor fusion algorithm for target tracking applications has been proposed based on the Student's t-distribution [32]. Note that the use of the Student's t-distribution raises significant difficulties of tractability and the variational Bayesian approach has been employed to derive approximated distributions. Nevertheless, the glint noise modeled by the Student's t-distribution has not been addressed for multi-target tracking in the RFS formulation.
In this paper, we attempt to apply the PHD filter to address the problem of multi-target tracking with glint noise. By modeling the glint noise as a Student's t-distribution, we show how the variational Bayesian approach can be used in the PHD filter to derive closed-form expressions. Based on the prior Gamma distributions for the parameters of the Student's t-distribution, we propose a novel implementation to the PHD filter by representing the predicted and the updated intensities as the mixtures of Gaussian–Gamma distributions. This is inspired by the idea of the GM-PHD filter for Gaussian distributions. To guarantee the same form of the predicted and the updated intensities, the main difficulty encountered is the computation of the predicted likelihood since the target state and the noise parameters are coupled in the likelihood functions. This is overcome by the variational Bayesian approximation method. In addition, a heuristic dynamics has been used which simply spreads the previous posterior distribution by a scalar factor so that the predicted intensity is still a mixture of Gaussian–Gamma terms. Simulation results are provided to illustrate the effectiveness of the proposed filter.
The rest of this paper is organized as follows. The problem of multi-target tracking with glint noise is formulated in Section 2. The implementation to the PHD filter is presented by applying the variational Bayesian approach in Section 3. In Section 4, a numerical example is provided to illustrate the effectiveness of the proposed filter. Conclusion is drawn in Section 5.
Section snippets
Tracking model
Consider the following linear target dynamics and measurement models:where and denote the target state and the measurement vectors, respectively. is the state transition matrix and Hk is the measurement matrix. The process noise is assumed to be zero-mean white Gaussian with covariance matrix and the measurement noise vk is assumed to be distributed according to heavy-tailed m-dimensional Student's t-distribution. Specifically, the
Main results
To derive a closed-form solution to the PHD recursion (8), (9), the intensities of the birth and spawning RFSs are assumed to be of the following forms:where , , , , , , and are given parameters that determine the shape of the birth intensity. , , , and
Simulation results
Consider a two-dimensional scenario with an unknown and time-varying number of targets. The target state is denoted by , where and denote the target position and velocity, respectively. The target dynamics is described by the nearly constant velocity modelwhere T=1 is the sampling period, and is zero-mean white Gaussian noise with covariance matrix
The range and the
Conclusion
In this paper, the PHD filter is applied for tracking an unknown and time-varying number of targets in glint noise environment. Based on a Student's t-distribution model for the glint noise statistics, the PHD filter is extended by augmenting the target state and the noise parameters and a novel implementation to the extended PHD filter is developed by using the variational Bayesian approach. The proposed filter is carried out by representing the predicted and the updated intensities as
Acknowledgments
This work was supported by the National Basic Research Program of China (973 Program, 2012CB821200, 2012CB821201), the NSFC (61203044, 61134005, 60921001, 90916024, 91116016) and the Beijing Natural Science Foundation (4132040).
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