PHD and CPHD Algorithms Based on a Novel Detection Probability Applied in an Active Sonar Tracking System

Tracking of midcourse ballistic target group on space-based infrared focal plane plays a key role in the space-based early warning system. This paper proposes the Gaussian-mixture cardinalized probability hypothesis density (GM-CPHD) filter to solve this problem. The multi-target states and measurements on infrared focal plane are modeled by random finite set (RFS). The intensity function of RFS of multi-target states and the probability distribution of target number are jointly propagated by cardinalized probability hypothesis density (CPHD) recursion. Under the assumptions of linear Gaussian multi-target models, the Gaussian-mixture implementations of CPHD are presented, and the target number and the multi-target states on infrared focal plane are estimated. In order to enable track continuity, we propose 0-1 integer programming to associate the estimated states between frames. The simulation results show that the GM-CPHD filter can dramatically improve the accuracy of estimated target number and estimated target states compared with the Gaussian-mixture probability hypothesis density filter, and that the track continuity can be successfully achieved. Defence Science Journal, 2012, 62(6), pp.431-436 , DOI:http://dx.doi.org/10.14429/dsj.62.1199

A distributed GM-CPHD filter based on parallel inverse covariance crossover is designed to attenuate the local filtering and uncertain time-varying noise affecting the accuracy of sensor signals. First, the GM-CPHD filter is identified as the module for subsystem filtering and estimation due to its high stability under Gaussian distribution. Second, the signals of each subsystem are fused by invoking the inverse covariance cross-fusion algorithm, and the convex optimization problem with high-dimensional weight coefficients is solved. At the same time, the algorithm reduces the burden of data computation, and data fusion time is saved. Finally, the GM-CPHD filter is added to the conventional ICI structure, and the generalization capability of the parallel inverse covariance intersection Gaussian mixture cardinalized probability hypothesis density (PICI-GM-CPHD) algorithm reduces the nonlinear complexity of the system. An experiment on the stability of Gaussian fusion models is organized and linear and nonlinear signals are compared by simulating the metrics of different algorithms, and the results show that the improved algorithm has a smaller metric OSPA error than other mainstream algorithms. Compared with other algorithms, the improved algorithm improves the signal processing accuracy and reduces the running time. The improved algorithm is practical and advanced in terms of multisensor data processing.

The probability hypothesis density (PHD) and cardinalized PHD (CPHD) filters were introduced in 2000 and 2006, respectively, as approximations of the full multitarget Bayes detection and tracking filter. Both filters are based on the standard multitarget measurement model that underlies most multitarget tracking theory. This paper is part of a series of theoretical studies that addresses PHD and CPHD filters for nonstandard multitarget measurement models. In this paper I derive the measurement-update equations for CPHD and PHD filters that estimate models of unknown, dynamically changing data, such as background clutter. A companion paper generalizes these results to multitarget detection and tracking in unknown, dynamic clutter.

The probability hypothesis density (PHD) and cardinalized PHD (CPHD) filters were introduced in 2000 and 2006, respectively, as approximations of the full multitarget Bayes detection and tracking filter. Both filters are based on the "standard" multitarget measurement model that underlies most multitarget tracking theory. This paper is part of a series of theoretical studies that addresses PHD and CPHD filters for nonstandard multitarget measurement models. In a companion paper I derived the measurement-update equations for CPHD and PHD filters for extracting clusters from dynamically evolving data sets. This paper uses these results to derive CPHD and PHD filters for detecting and tracking multiple targets obscured by unknown, dynamically changing clutter.

Asynchronism among the local radars is one of the most important challenges for multi-target tracking (MTT) in multi-static radar systems. In order to address this challenge in the framework of random finite set (RFS) based on Bayesian inference, we propose a decentralized probability hypothesis density (PHD) filter based on the asynchronous periodical sequential estimation (APSE). First, starting from a multi-target Bayesian filter, we derive the multi-target density update expressions for the APSE solution in the RFS framework. Next, we develop the PHD recursion expressions of the APSE solution, named as APSE-PHD, and describe the Gaussian mixture (GM) implementation of the APSE-PHD. Simulation results for a challenging tracking scenario confirm that the proposed APSE-PHD algorithm is effective for MTT in the asynchronous multistatic radar system and outperforms the existing PHD-based algorithm.

Multitarget tracking in the interference environments suffers from the nonuniform, unknown, and time-varying clutter, resulting in dramatic performance deterioration. We address this challenge by proposing a robust multitarget tracking algorithm, which estimates the states of clutter and targets simultaneously by the message-passing (MP) approach. We define the nonhomogeneous clutter with a finite mixture model containing a uniform component and multiple nonuniform components. The measured signal strength is utilized to estimate the mean signal-to-noise ratio of targets and the mean clutter-to-noise ratio of clutter, which are then used as additional feature information of targets and clutter to improve the performance of discrimination of targets from clutter. We also present a hybrid data association, which can reason over correspondence between targets, clutter, and measurements. Then, a unified MP algorithm is used to infer the marginal posterior probability distributions of targets, clutter, and data association by splitting the joint probability distribution into a mean-field approximate part and a belief propagation part. As a result, a closed-loop iterative optimization of the posterior probability distribution can be obtained, which can effectively deal with the coupling between target tracking, clutter estimation, and data association. Simulation results demonstrate the performance superiority and robustness of the proposed multitarget tracking algorithm compared with the probability hypothesis density (PHD) filter and the cardinalized PHD (CPHD) filter.

Different from the standard probability hypothesis density (PHD) and cardinality probability hypothesis density (CPHD) filters, the trajectory PHD (TPHD) and trajectory CPHD (TCPHD) filters employ the sets of trajectories rather than the sets of the targets as the variables for multi-target filtering. The TPHD and TCPHD filters exploit the inherent potential of the standard PHD and CPHD filters to generate the target trajectory estimates from first principles. In this paper, we develop the TPHD and TCPHD filters for pulse Doppler radars (PD-TPHD and PD-TCPHD filters) to improve the multi-target tracking performance in the scenario with clutter. The Doppler radar can obtain the Doppler measurements of targets in addition to the position measurements of targets, and both measurements are integrated into the recursive filtering of PD-TPHD and PD-TCPHD. PD-TPHD and PD-TCPHD can propagate the best augmented Poisson and independent identically distributed multi-trajectory density approximation, respectively, through the Kullback–Leibler divergence minimization operation. Considering the low computational complexity of sequential filtering, Doppler measurements are sequentially applied to the Gaussian mixture implementation. Moreover, we perform the L-scan implementations of PD-TPHD and PD-TCPHD. Simulation results demonstrate the effectiveness and robustness of the proposed algorithms in the scenario with clutter.

Multi-target tracking in distributed sensor networks using particle PHD filters

Tracking individuals within a group is one of the major tasks of group target observation. Tracking radar must feature high frame rate and high range-angular resolution to achieve the stable multitarget tracking performance. However, two major problems arise from this scenario. First, the narrow beam of tracking radar does not allow the complete observation of group target, causing the fluctuation of target number as the radar-target geometry changes; second, false alarms may be target-dependent and distributed around the targets, which is contrary to the traditional spatially uniform clutter model. This paper proposes a Gaussian mixture probability hypothesis density filter for multitarget tracking using a collaborative radar system. The system consists of one scanning radar and one tracking radar. The former outputs the group's collective states (centroid, extension, etc.), which are used as the priors for the tracking radar. The tracking radar is responsible for the multitarget tracking. The density of target birth and death are set according to the priors. The update equation of the probability hypothesis density in target-dependent false alarms is derived and simplified to meet the practical application requirements. Finally, the effectiveness of the proposed filter is verified by the simulation and experimental results.

The probability hypothesis density (PHD) recursion is a first moment approximation to the multi-target Bayes filter which propagates the posterior intensity of the random finite set of targets in time. The cardinalized PHD (CPHD) recursion is a generalization of the PHD recursion, which jointly propagates the posterior intensity and the posterior cardinality distribution. Using the recently developed closed form solutions to both the PHD and CPHD recursions for linear Gaussian multi-target models, we present a comparative study of their performances

Defining a multi-target motion model, which is an important step of tracking algorithms, can be very challenging. Using fixed models (as in several generative Bayesian algorithms, such as Kalman filters) can fail to accurately predict sophisticated target motions. On the other hand, sequential learning of the motion model (for example, using recurrent neural networks) can be computationally complex and difficult due to the variable unknown number of targets. In this paper, we propose a multi-target filtering and tracking algorithm which learns the motion model, simultaneously for all targets, from an implicitly represented state map and performs spatio-temporal data prediction. To this end, the multi-target state is modelled over a continuous hypothetical target space, using random finite sets and Gaussian mixture probability hypothesis density formulations. The prediction step is recursively performed using a deep convolutional recurrent neural network with a long short-term memory architecture, which is trained as a regression block, on the fly, over "probability density difference" maps. Our approach is evaluated over widely used pedestrian tracking benchmarks, remarkably outperforming state-of-the-art multi-target filtering algorithms, while giving competitive results when compared with other tracking approaches: The proposed approach generates an average 40.40 and 62.29 optimal sub-pattern assignment (OSPA) errors on MOT15 and MOT16/17 datasets, respectively, while producing 62.0%, 70.0% and 66.9% multi-object tracking accuracy (MOTA) on MOT16/17, PNNL Parking Lot and PETS09 pedestrian tracking datasets, respectively, when publicly available detectors are used.

The probability hypothesis density (PHD) filter is a practical alternative to the optimal Bayesian multi-target filter based on random finite sets. It propagates the PHD function, the first order moment of the posterior multi-target density, from which the number of targets as well as their individual states can be extracted. Furthermore, the sequential Monte Carlo (SMC) approximation of the PHD filter (also known as particle-PHD filter) is available in the literature in order to overcome its intractability. However, the PHD filter keeps no track of the target identities and hence cannot produce track-valued estimates of individual targets. This work consider the use of an improved implementation, of the particle-PHD filter that gives the track-valued estimates of individual targets and propose a novel way for doing so. The improved PHD filter combines the particles approximation of the posterior PHD function and the peak extraction from the posterior PHD particles to create the target identities of the individual estimates. The improved PHD filter does not affect the convergence results of the particle-PHD filter

Refined GM-PHD tracker for tracking targets in possible subsequent missed detections

In this paper, we provide novel derivations of the probability hypothesis density (PHD) and cardinalised PHD (CPHD) filters without using probability generating functionals or functional derivatives. We show that both the PHD and CPHD filters fit in the context of assumed density filtering and implicitly perform Kullback–Leibler divergence (KLD) minimizations after the prediction and update steps. We perform the KLD minimizations directly on the multitarget prediction and posterior densities.

The single sensor probability hypothesis density (PHD) and cardinalized probability hypothesis density (CPHD) filters have been developed in the literature using the random finite set framework. The existing multisensor extensions of these filters have limitations such as sensor order dependence, numerical instability or high computational requirements. In this paper we derive update equations for the multisensor CPHD filter. The multisensor PHD filter is derived as a special case. Exact implementation of the multisensor CPHD involves sums over all partitions of the measurements from different sensors and is thus intractable. We propose a computationally tractable approximation which combines a greedy measurement partitioning algorithm with the Gaussian mixture representation of the PHD. Our greedy approximation method allows the user to control the tradeoff between computational overhead and approximation accuracy.