A Gaussian Mixture PHD Filter for Jump Markov System Models

The probability hypothesis density (PHD) filter is an attractive approach to tracking an unknown and time-varying number of targets in the presence of data association uncertainty, clutter, noise, and detection uncertainty. The PHD filter admits a closed-form solution for a linear Gaussian multi-target model. However, this model is not general enough to accommodate maneuvering targets that switch between several models. In this paper, we generalize the notion of linear jump Markov systems to the multiple target case to accommodate births, deaths, and switching dynamics. We then derive a closed-form solution to the PHD recursion for the proposed linear Gaussian jump Markov multi-target model. Based on this an efficient method for tracking multiple maneuvering targets that switch between a set of linear Gaussian models is developed. An analytic implementation of the PHD filter using statistical linear regression technique is also proposed for targets that switch between a set of nonlinear models. We demonstrate through simulations that the proposed PHD filters are effective in tracking multiple maneuvering targets.     

交互多模型PHD滤波跟踪多目标方法

在机动多目标跟踪问题中,目标数未知或随时间而变化,概率假设密度（PHD）滤波可以在每一时间步估计多目标状态和目标数,但单模型方法不能给出精确的估计。提出了一种交互多模型PHD滤波方法,建立多模型描述多目标运动方式,利用PHD滤波结合多模型跟踪目标运动轨迹。同时,给出了多传感器交互多模型PHD滤波方法,以提高目标跟踪精度。     

多目标跟踪的概率假设密度粒子滤波

在多目标跟踪中,当目标数很大时,目标状态的联合分布的计算量会非常大。如果目标独立运动,可用各目标分别滤波来代替,但这要求考虑数据互联问题。文章介绍一种可以解决计算量问题的方法,只需计算联合分布的一阶矩——概率假设密度(PHD),PHD在任意区域S上的积分是S内目标数的期望值。因未记录目标身份,避免了数据互联问题。仿真中,传感器为被动雷达,目标观测值为距离、角度及速度时,对上述的PHD滤波进行了粒子实现,并对观测值是否相关的不同情况进行比较。PHD粒子滤波应用在非线性模型的多目标跟踪,实验结果表明,滤波可以稳健跟踪目标数为变数的情况,得到了接近真实情况的结果。     

Multitarget Bayes filtering via first-order multitarget moments

The theoretically optimal approach to multisensor-multitarget detection, tracking, and identification is a suitable generalization of the recursive Bayes nonlinear filter. Even in single-target problems, this optimal filter is so computationally challenging that it must usually be approximated. Consequently, multitarget Bayes filtering will never be of practical interest without the development of drastic but principled approximation strategies. In single-target problems, the computationally fastest approximate filtering approach is the constant-gain Kalman filter. This filter propagates a first-order statistical moment - the posterior expectation - in the place of the posterior distribution. The purpose of this paper is to propose an analogous strategy for multitarget systems: propagation of a first-order statistical moment of the multitarget posterior. This moment, the probability hypothesis density (PHD), is the function whose integral in any region of state space is the expected number of targets in that region. We derive recursive Bayes filter equations for the PHD that account for multiple sensors, nonconstant probability of detection, Poisson false alarms, and appearance, spawning, and disappearance of targets. We also show that the PHD is a best-fit approximation of the multitarget posterior in an information-theoretic sense.     

Track labeling and PHD filter for multitarget tracking

Multiple target tracking requires data association that operates in conjunction with filtering. When multiple targets are closely spaced, the conventional approaches (as, e.g., MHT/assignment) may not give satisfactory results. This is mainly because of the difficulty in deciding what the number of targets is. Recently, the probability hypothesis density (PHD) filter has been proposed and particle filtering techniques have been developed to implement the PHD filter. In the particle PHD filter, the track labeling problem is not considered, i.e., the PHD is obtained only for a frame at a time, and it is very difficult to perform the multipeak extraction, particularly in high clutter environments. A track labeling method combined with the PHD approach, as well as considering the finite resolution, is proposed here for multitarget tracking, i.e., we keep a separate tracker for each target, use the PHD in the resolution cell to get the estimated number and locations of the targets at each time step, and then perform the track labeling ("peak-to-track" association), whose results can provide information for PHD peak extraction at the next time step. Besides, by keeping a separate tracker for each target, our approach provides more information than the standard particle PHD filter. For example, in group target tracking, if we are interested in the motion of a specific target, we can track this target, which is not possible for the standard particle PHD filter, since the standard particle PHD filter does not keep track labels. Using our approach, multitarget tracking can be performed with automatic track initiation, maintenance, spawning, merging, and termination     

Novel data association schemes for the probability hypothesis density filter

The probability hypothesis density (PHD) filter is a practical alternative to the optimal Bayesian multi-target Alter based on finite set statistics. It propagates the PHD function, a first-order moment of the full multi-target posterior density. The peaks of the PHD function give estimates of target states. However, the PHD filter keeps no record of target identities and hence does not produce track-valued estimates of individual targets. We propose two different schemes according to which PHD filter can provide track-valued estimates of individual targets. Both schemes use the probabilistic data-association functionality albeit in different ways. In the first scheme, the outputs of the PHD filter are partitioned into tracks by performing track-to-estimate association. The second scheme uses the PHD filter as a clutter filter to eliminate some of the clutter from the measurement set before it is subjected to existing data association techniques. In both schemes, the PHD filter effectively reduces the size of the data that would be subject to data association. We consider the use of multiple hypothesis tracking (MHT) for the purpose of data association. The performance of the proposed schemes are discussed and compared with that of MHT.     

An improved multiple model GM-PHD filter for maneuvering target tracking

In this paper, an improved implementation of multiple model Gaussian mixture probability hypothesis density (MM-GM-PHD) filter is proposed. For maneuvering target tracking, based on joint distribution, the existing MM-GM-PHD filter is relatively complex. To simplify the filter, model conditioned distribution and model probability are used in the improved MM-GM-PHD filter. In the algorithm, every Gaussian components describing existing, birth and spawned targets are estimated by multiple model method. The final results of the Gaussian components are the fusion of multiple model estimations. The algorithm does not need to compute the joint PHD distribution and has a simpler computation procedure. Compared with single model GM-PHD, the algorithm gives more accurate estimation on the number and state of the targets. Compared with the existing MM-GM-PHD algorithm, it saves computation time by more than 30%. Moreover, it also outperforms the interacting multiple model joint probabilistic data association (IMMJPDA) filter in a relatively dense clutter environment.     

PHD filters of higher order in target number

The multitarget recursive Bayes nonlinear filter is the theoretically optimal approach to multisensor-multitarget detection, tracking, and identification. For applications in which this filter is appropriate, it is likely to be tractable for only a small number of targets. In earlier papers we derived closed-form equations for an approximation of this filter based on propagation of a first-order multitarget moment called the probability hypothesis density (PHD). In a recent paper, Erdinc, Willett, and Bar-Shalom argued for the need for a PHD-type filter which remains first-order in the states of individual targets, but which is higher-order in target number. In this paper we show that this is indeed possible. We derive a closed-form cardinalized PHD (CPHD) filter, which propagates not only the PHD but also the entire probability distribution on target number.     

Sequential Monte Carlo methods for multitarget filtering with random finite sets

Random finite sets (RFSs) are natural representations of multitarget states and observations that allow multisensor multitarget filtering to fit in the unifying random set framework for data fusion. Although the foundation has been established in the form of finite set statistics (FISST), its relationship to conventional probability is not clear. Furthermore, optimal Bayesian multitarget filtering is not yet practical due to the inherent computational hurdle. Even the probability hypothesis density (PHD) filter, which propagates only the first moment (or PHD) instead of the full multitarget posterior, still involves multiple integrals with no closed forms in general. This article establishes the relationship between FISST and conventional probability that leads to the development of a sequential Monte Carlo (SMC) multitarget filter. In addition, an SMC implementation of the PHD filter is proposed and demonstrated on a number of simulated scenarios. Both of the proposed filters are suitable for problems involving nonlinear nonGaussian dynamics. Convergence results for these filters are also established.     

Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets

The probability hypothesis density (PHD) filter has been recognized as a promising tech-nique for tracking an unknown number of targets. The performance of the PHD filter, however, is sensitive to the available knowledge on model parameters such as the measurement noise variance and those associated with the changes in the maneuvering target trajectories. If these parameters are unknown in advance, the tracking performance may degrade greatly. To address this aspect, this paper proposes to incorporate the adaptive parameter estimation (APE) method in the PHD filter so that the model parameters, which may be static and/or time-varying, can be estimated jointly with target states. The resulting APE-PHD algorithm is implemented using the particle filter (PF), which leads to the PF-APE-PHD filter. Simulations show that the newly proposed algorithm can correctly identify the unknown measurement noise variances, and it is capable of tracking mul-tiple maneuvering targets with abrupt changing parameters in a more robust manner, compared to the multi-model approaches.     

Efficient particle filtering for road-constrained target tracking

The variable-structure multiple-model particle filtering approach for state estimation of road-constrained targets is addressed. The multiple models are designed to account for target maneuvers including "move-stop-move" and motion ambiguity at an intersection; the time-varying active model sets are adaptively selected based on target state and local terrain condition. The hybrid state space is partitioned into the mode subspace and the target subspace. The mode state is estimated based on random sampling; the target state as well as the relevant likelihood function associated with a mode sample sequence is approximated as Gaussian distribution, of which the conditional mean and covariance are deterministically computed using a nonlinear Kalman filter which accounts for road constraints in its update. The importance function for the sampling of the mode state approximates the optimal importance function under the same Gaussian assumption of the target state.     

A measurement-driven adaptive probability hypothesis density filter for multitarget tracking

This paper studies the dynamic estimation problem for multitarget tracking. A novel gating strategy that is based on the measurement likelihood of the target state space is proposed to improve the overall effectiveness of the probability hypothesis density(PHD) filter. Firstly, a measurement-driven mechanism based on this gating technique is designed to classify the measurements. In this mechanism, only the measurements for the existing targets are considered in the update step of the existing targets while the measurements of newborn targets are used for exploring newborn targets. Secondly, the gating strategy enables the development of a heuristic state estimation algorithm when sequential Monte Carlo(SMC) implementation of the PHD filter is investigated, where the measurements are used to drive the particle clustering within the space gate.The resulting PHD filter can achieve a more robust and accurate estimation of the existing targets by reducing the interference from clutter. Moreover, the target birth intensity can be adaptive to detect newborn targets, which is in accordance with the birth measurements. Simulation results demonstrate the computational efficiency and tracking performance of the proposed algorithm.     

Rao-Blackwellized粒子概率假设密度滤波算法

针对多目标跟踪（MTT）,提出一种新的基于随机集的滤波算法,称为Rao-Blackwellized粒子概率假设密度滤波算法（RBP-PHDF）。算法运用Rao-Blackwellized思想,通过挖掘分析“混合线性/非线性模型”的结构,采用序列蒙特卡罗（SMC）方法预测与估计概率假设密度（PHD）迭代式中各个目标的非线性状态,并利用非线性状态粒子中包含的信息,使用卡尔曼滤波器（KF）对线性状态进行预测与估计。以更好地估计PHD进而提高各目标状态估计精度。分析与MTT仿真的结果表明,在相同的仿真条件下,与现有序列蒙特卡罗概率假设密度滤波算法（SMC-PHDF）相比,RBP-PHDF在降低粒子维数、减少计算量的同时,有效提升了估计精度。     

Multiple model nonlinear filtering for low signal ground target applications

The design and implementation of a multiple model nonlinear filter (MMNLF) for ground target tracking using ground moving target indicator (GMTI) radar measurements is described. Like the well-known interacting multiple model Kalman filter (IMMKF), the MMNLF is based on the theory of hybrid stochastic systems. However, since it models the probability distribution for the target in a region, rather than just the distribution's first and second moments, a nonlinear filter is able to capture more fine-grained detail of the target motion and requires fewer models than typical IMMKF implementations. This is illustrated here with a two-model MMNLF in which one motion model incorporates terrain constraints while the second is a nearly constant velocity (CV) model. Another feature of the MMNLF is that it enables incorporation of prethresholded measurements. To implement the filter, the target state conditional probability density is discretized on a set of moving grids and recursively updated with sensor measurements via Bayes' formula. The conditional density is time updated between sensor measurements using alternating direction implicit (ADI) finite difference methods, generalized for this hybrid application. In simulation testing against low signal-to-interference-plus-noise ratio (SINR) targets, the MMNLF is able to maintain track in situations where single model filters based on either of the component models or filters that use thresholded data fail. Potential applications of this work include detection and tracking of foliage-obscured moving targets.     

Precision large scale air traffic surveillance using IMM/assignmentestimators

We present the development and implementation of a multisensor-multitarget tracking algorithm for large scale air traffic surveillance based on interacting multiple model (IMM) state estimation combined with a 2-dimensional assignment for data association. The algorithm can be used to track a large number of targets from measurements obtained with a large number of radars. The use of the algorithm is illustrated on measurements obtained from 5 FAA radars, which are asynchronous, heterogeneous, and geographically distributed over a large area. Both secondary radar data (beacon returns from cooperative targets) as well as primary radar data (skin returns from noncooperative targets) are used. The target IDs from the beacon returns are not used in the data association. The surveillance region includes about 800 targets that exhibit different types of motion. The performance of an IMM estimator with linear motion models is compared with that of the Kalman filter (KF). A number of performance measures that can be used on real data without knowledge of the ground truth are presented for this purpose. It is shown that the IMM estimator performs better than the KF. The advantage of fusing multisensor data is quantified. It is also shown that the computational requirements in the multisensor case are lower than in single sensor case, Finally, an IMM estimator with a nonlinear motion model (coordinated turn) is shown to further improve the performance during the maneuvering periods over the IMM with linear models     

Adaptive Tracker Field-of-View Variation Via Multiple Model Filtering

Adaptive estimation using multiple model filtering is investigated as a means of changing the field of view as well as the bandwidth of an infrared image tracker when target acceleration can vary over a wide range. The multiple models are created by tuning filters for best performance at differing conditions of exhibited target behavior and differing physical size of their respective fields of view. Probabilistically weighted averaging provides the adaptation mechanism. Each filter involves online identification of the target shape function, so that this algorithm can be used against ill-defined and/or multiple-hot-spot targets. When each individual filter has the form of an enhanced correlator/linear Kalman filter, computational loading is very low. In contrast, an extended Kalman filter processing the raw infrared data directly and assuming a nonlinear constant turn-rate dynamics model provides superior tracking capability, especially for harsh maneuvers, at the cost of a larger computational burden.     

改进的MMPHD机动目标跟踪方法

基于序列蒙特卡罗方法的经典多模概率假设密度滤波方法及其各种衍生方法,在预测过程中依据多个并行的状态转移模型,通过将大量粒子散布到下一时刻目标所有可能出现的状态空间实现目标状态的捕获,造成计算量大、目标跟踪精度差。为此,提出一种改进的多模粒子概率假设密度机动目标跟踪方法。该方法利用最新量测信息估计目标运动模型概率及模型参数,并将估计得到的目标模型应用到粒子概率假设密度滤波方法的预测过程中生成预测粒子,从而将大部分粒子聚合在目标最可能出现的状态空间邻域中,实现粒子的有效利用。数值仿真表明,所提方法不仅显著地减少了目标丢失个数,而且提高了目标跟踪精度。     

基于标签多伯努利滤波的衍生目标跟踪算法

针对现有随机有限集(RFS)滤波器在低信噪比环境下对衍生目标跟踪性能严重下降的问题,提出了一种基于Delta扩展标签多伯努利(δ-GLMB)滤波器的改进算法。基于随机集理论和伯努利衍生模型,推导了新的预测方程,并采用了假设裁剪及分组手段和多伯努利近似技术以降低算法的计算量。针对假设增多引起的虚警问题,将多帧平滑思想和算法相结合,利用标签信息对新目标进行回溯处理。仿真结果表明,所提算法能对目标数目进行无偏估计,在低探测概率和强杂波环境下性能明显优于概率假设密度(PHD)算法,计算开销在衍生初始阶段增长快于PHD,目标较分散时低于PHD。     

Multitarget tracking using the joint multitarget probability density

This work addresses the problem of tracking multiple moving targets by recursively estimating the joint multitarget probability density (JMPD). Estimation of the JMPD is done in a Bayesian framework and provides a method for tracking multiple targets which allows nonlinear target motion and measurement to state coupling as well as nonGaussian target state densities. The JMPD technique simultaneously estimates both the target states and the number of targets in the surveillance region based on the set of measurements made. We give an implementation of the JMPD method based on particle filtering techniques and provide an adaptive sampling scheme which explicitly models the multitarget nature of the problem. We show that this implementation of the JMPD technique provides a natural way to track a collection of targets, is computationally tractable, and performs well under difficult conditions such as target crossing, convoy movement, and low measurement signal-to-noise ratio (SNR).     

Data Association and Track Management for the Gaussian Mixture Probability Hypothesis Density Filter

The Gaussian mixture probability hypothesis density (GM-PHD) recursion is a closed-form solution to the probability hypothesis density (PHD) recursion, which was proposed for jointly estimating the time-varying number of targets and their states from a sequence of noisy measurement sets in the presence of data association uncertainty, clutter, and miss-detection. However the GM-PHD filter does not provide identities of individual target state estimates, that are needed to construct tracks of individual targets. In this paper, we propose a new multi-target tracker based on the GM-PHD filter, which gives the association amongst state estimates of targets over time and provides track labels. Various issues regarding initiating, propagating and terminating tracks are discussed. Furthermore, we also propose a technique for resolving identities of targets in close proximity, which the PHD filter is unable to do on its own.