We address the classical bearings-only tracking problem (BOT) for a single object, which belongs to the general class of nonlinear filtering problems. Recently, algorithms based on sequential Monte-Carlo methods (particle filtering) have been proposed. As far as performance analysis is concerned, the posterior Cramer-Rao bound (PCRB) provides a lower bound on the mean square error. Classically, under a technical assumption named "asymptotic unbiasedness assumption," the PCRB is given by the inverse Fisher informatiom matrix (FIM). The lalter is computed using Tichavsky's recursive formula via Monte-Carlo methods. Two major problems are studied here. First, we show that the asymptotic unbiasedness assumption can be replaced by an assumption which is more meaningful. Second, an exact algorithm to compute the PCRB is derived via Tichavský's recnrsive formula without using Monte-Carlo methods. This result is based on a new coordinate system named logarithmic polar coordinate (LPC) system. Simulation results illustrate that PCRB can now be computed accurately and quickly, making it suitable for sensor management applications.

Closed-Form Posterior Cramér-Rao Bounds for Bearings-Only Tracking

The Probability Hypothesis Density (PHD) filter is a multitarget tracker that can alleviate the computational intractability of the optimal multitarget Bayes filter. The PHD filter recursively estimates the number of targets and their PHD from a set of observations and works well in scenarios with false alarms and missed detections. Two distinct PHD filter implementations are available in the literature: the Sequential Monte Carlo Probability Hypothesis Density (SMC-PHD) and the Gaussian Mixture Probability Hypothesis Density (GM-PHD) filters. While particle-based PHD implementations may suffer from degeneracy, GM-based methods may not be suitable for highly nonlinear non-Gaussian systems. This paper proposes a B-Spline based Spline Probability Hypothesis Density (SPHD) filter, which has the capability to better approximate any arbitrary probability density function. The resulting algorithm can handle linear, non-linear, Gaussian, and non-Gaussian models. The SPHD filter can provide continuous estimates of the probability density function of the system state and it is immune to the degeneracy problem. The SPHD filter can maintain highly accurate tracks by taking advantage of dynamic knot movement, but at the expense of higher computational complexity, which makes it suitable for tracking a few high-value targets under difficult conditions. The SPHD filter derivations and simulations are provided in this paper.

The Spline Probability Hypothesis Density Filter

A two-filter based method of detecting and tracking a target that can track an unknown and time-varying number of targets, while keeping continuous track, even in scenarios with large number of false contacts or missing measurements, is provided. More specifically, a first filter provides target detection, a second filter provides target tracking of the detected targets, and a clustering technique that operates after the first filter. The first filter starts with a uniform distribution over the surveillance area and resets periodically after the clustering technique is run. When the clustering technique runs, it detects the clusters corresponding to the different targets and passes them to the second filter that tracks these targets.

Two-stage filtering based method for multiple target tracking

Intensive business competition has encouraged enterprises to focus more on their core activities to outsource the management of supply chain. Issues such as political crises, demand fluctuations, strategy variations, financial instability and natural disasters are the factors that can contribute to emergence of uncertainty and risk in the supply chain. Effective management of such risks is essential for reducing the vulnerability of the supply chain. Supply risk management is the most basic concept of the supply chain risk management; therefore, this study examines this subject by developing a mathematical model consisting of a set of equations to quantify typical risk such as delayed delivery, substandard quality, natural disasters and financial risk of the supplier. This model is designed to select and allocate orders to suppliers to minimize the supply risk imposed by the mentioned risk factors. A fuzzy-order function is implemented to turn this crisp model into a fuzzy supply chain risk assessment model. The defuzzified solutions of the fuzzy model show significant improvements over the result of crisp model. Uncertainty in supply chain and intense competitiveness between organizations and managers create different challenges. To effectively manage these challenges, new management approaches are used to strengthen and improve the effectiveness of organizations. The results show that the fuzzy model yields better results compared to the deterministic model. Moreover, it is demonstrated that the ability of the model for activities risk control is improved. The work presented addresses the same challenging problems that others have tried to address, but the methodology presented in this paper is more efficient in terms of execution performance and accuracy.

Implementing fuzzy rank function model for a new supply chain risk management

We propose an efficient sequential Monte Carlo probability hypothesis density (PHD) filter which employs the Kalman-gain approach during weight update to correct predicted particle states by minimizing the mean square error between the estimated measurement and the actual measurement received at a given time in order to arrive at a more accurate posterior. This technique identifies and selects those particles belonging to a particular target from a given PHD for state correction during weight computation. Besides the improved tracking accuracy, fewer particles are required in the proposed approach. Simulation results confirm the improved tracking performance when evaluated with different measures.

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Kalman-Gain Aided Particle PHD Filter for Multitarget Tracking

In this article, the classical bearings-only tracking (BOT) problem for a single target is addressed, which belongs to the general class of non-linear filtering problems. Due to the fact that the radial distance observability of the target is poor, the algorithm-based sequential Monte-Carlo (particle filtering, PF) methods generally show instability and filter divergence. A new stable distributed multi-sensor PF method is proposed for BOT. The sensors process their measurements at their sites using a hierarchical PF approach, which transforms the BOT problem from Cartesian coordinate to the logarithmic polar coordinate and separates the observable components from the unobservable components of the target. In the fusion centre, the target state can be estimated by utilising the multi-sensor optimal information fusion rule. Furthermore, the computation of a theoretical Cramer–Rao lower bound is given for the multi-sensor BOT problem. Simulation results illustrate that the proposed tracking method can provid...

Distributed multi-sensor particle filter for bearings-only tracking

When the number of targets is unknown or varies with time, multitarget state and measurements are represented as random sets and the multitarget tracking problem is addressed by calculating the first moment of the joint distribution, the probability hypothesis density (PHD), recursively. The PHD filter has been generalized to the cardinalized PHD (CPHD) filter, which propagates not only the PHD but also the full probability distribution on target number. In this paper, a new particle CPHD filter in the Gaussian mixture framework is presented. The CPHD filter uses a bank of Gaussian particle filters (GPFs) to approximate each Gaussian component and does not require clustering to determine target states. Moreover, Quasi-Monte Carlo (QMC) integration method is introduced to approximating the prediction and update distributions of target states. A challenging passive bearings-only multitarget tracking scenario is used to evaluate the performance of the proposed CPHD filter.

A new Gaussian mixture particle CPHD filter for multitarget tracking

In this paper, a novel multitarget bearings-only tracking algorithm that combines the fuzzy clustering data association technique together with a Gaussian particle filter (GPF) is presented. Firstly, to deal with the data association problem that arises due to the uncertainty of the measurements, the fuzzy clustering method with the maximum entropy principle is utilized, which eliminates those invalid measurements. Secondly, this paper employs GPF to update each target state independently, since it has a much-improved performance and versatility over other Gaussian filters, especially when nontrivial nonlinearities are present. Moreover, in the multisensor scenario, a statistic test method based on the cotangent values of bearings is proposed, for associating the target bearing data observed at each sensor. Simulation results demonstrate the effectiveness of the algorithm.

Multitarget bearings-only tracking using fuzzy clustering technique and Gaussian particle filter

A novel multitarget tracking algorithm that combines the maximum entropy fuzzy (MEF) clustering data association technique together with Gaussian particle filter (GPF) is presented. Firstly, the MEF clustering approach is provided to deal with the data association problem that arises due to the uncertainty of the measurements, which eliminates those invalidate measurements. Since GPF has much-improved performance and versatility over other Gaussian filters, especially when nontrivial nonlinearities are present, this paper employs it and joint association innovations to update each target state independently. Finally, the proposed algorithm is applied to multitarget bearings-only tracking. Simulation results demonstrate the effectiveness of the algorithm.

A Novel Multitarget Tracking Algorithm Based on Fuzzy Clustering Technique and Gaussian Particle Filter

For bearings-only tracking (BOT), there are mainly two problems of nonlinear filtering and poor range observability. In the paper, a new distributed multisensor pseudolinear Kalman filter (PLKF) algorithm is proposed. The sensors use an instrumental vector PLKF (IV-PLKF) to process the measurements of the target independently, which can tackle the bias arising from the correlation between the measurement vector and pseudolinear noise by the bias compensation PLKF (BC-PLKF). The IV-PLKF embeds the recursive instrumental vector estimation method into the BC-PLKF, uses it to construct the instrumental vector, and applies the method of selective angle measurement to modify the local target state estimation and covariance. In the fusion center, the target state can be estimated by using the multisensor optimal information fusion criterion. Then the Cramer-Rao lower bound (CRLB) of multisensor BOT is derived. Simulation results show the effectiveness of the algorithm.

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Jungen Zhang

Papers

Distributed multi-sensor particle filter for bearings-only tracking

A new Gaussian mixture particle CPHD filter for multitarget tracking

Multitarget bearings-only tracking using fuzzy clustering technique and Gaussian particle filter

A Novel Multitarget Tracking Algorithm Based on Fuzzy Clustering Technique and Gaussian Particle Filter

Bearings-only Tracking Based on Distributed Multisensor Pseudolinear Kalman Filter