Cardinalized Probability Hypothesis Density Filter Research Articles

A Gaussian Mixture CPHD Filter for Multi-Target Tracking in Target-Dependent False Alarms

The estimation of the target number and individual tracks are two major tasks in multi-target tracking. The main shortcoming of traditional tracking methods is the cumbersome data association between measurements and targets. The cardinalized probability hypothesis density filter (CPHD) proposed in recent years can achieve the requirement for multitarget tracking. This kind of filter jointly estimates the cardinality distribution and the posterior density, which can achieve a more stable estimate of the target number. However, targets with complex micro-Doppler signatures (drones, birds, etc.) may generate target-dependent false alarms, which is contrary to the traditional uniform distribution assumption. In this case, the estimates of traditional CPHD filter will suffer from the abnormal transfer of PHD mass, causing the degradation of filtering performance. This paper studies the individual tracking of group targets with an improved GM-CPHD filter. First, the target-dependent false alarms are modeled with a general independent and identically distributed (I.I.D.) cluster process. Second, the update equations of cardinality and PHD density in target-dependent false alarms are derived. Finally, a practical solution using the Gaussian mixture method is proposed. The effectiveness of the proposed filter is verified by the simulation and experimental results.

Robust cardinalized probability hypothesis density filter based underwater multi-target direction-of-arrival tracking with uncertain measurement noise

Direction-of-arrival (DOA) tracking of underwater targets is an important research topic in sonar signal processing. Considering the underwater DOA tracking is a typical multi-target scenario with uncertain measurement noise caused by unknown underwater environment, a robust underwater multi-target DOA tracking method with uncertain measurement noise is proposed by implementing a spatial matrix filtering (SMF) technique in the framework of the cardinalized probability hypothesis density (CPHD) filter. Firstly, the kinematic model of underwater targets and the measurement model based on the received signal of the hydrophone array (array signal) are established. Thus, the multi-target DOA tracking algorithm is derived by using the CPHD filter based on the established model. Moreover, to avoid the effect of the uncertain measurement noise, the SMF technique is implemented to process the array signal to extract the signal in the bearing angle of the target. Then, the output of the SMF is substituted into the CPHD filter. In this manner, the robust underwater multi-target DOA tracking method for uncertain measurement noise scenario is proposed. Finally, comprehensive simulations and experimental data processing are performed to prove the robustness of the proposed method in the scenario of multi-target DOA tracking with uncertain measurement noise. Compared with the existing DOA estimation and DOA tracking methods, the proposed methods significantly improve the robustness and the precision of the DOA of the underwater target.

Variational Bayesian cardinalized probability hypothesis density filter for robust underwater multi-target direction-of-arrival tracking with uncertain measurement noise

The direction-of-arrival (DOA) tracking of underwater targets is an important research topic in sonar signal processing. Considering that the underwater DOA tracking is a typical multi-target problem under unknown underwater environment with missing detection, false alarm, and uncertain measurement noise, a robust underwater multi-target DOA tracking method for uncertain measurement noise is proposed. First, a kinematic model of the multiple underwater targets and bearing angle measurement model with missing detection and false alarms are established. Then, the multi-target DOA tracking algorithm is derived by using the cardinalized probability hypothesis density (CPHD) filter, the performance of which largely depends on the accuracy of the parameter of measurement noise variance. In addition, the variational Bayesian approach is used to adaptively estimate the uncertain measurement of noise variance for each measurement of target in the real time of tracking. Thus, the robust underwater multi-target DOA tracking is carried out. Finally, comprehensive experimental validations and discussions are made to prove that the proposed algorithm can provide robust DOA tracking in the multi-target tracking scenario with uncertain measurement noise.

A general cardinalized probability hypothesis density filter

Based on random finite set, the probability hypothesis density (PHD) filter and the cardinalized PHD (CPHD) filter have been proposed for multitarget tracking as they are computational tractable. But the classical PHD and CPHD filter is not applicable when a target generates multiple detections. The general PHD filter was proposed to apply the PHD filter to arbitrary clutter and target measurement process. However, as a filter with better performance than the PHD filter, the CPHD filter for multiple-detection tracking is not proposed except the extended target CPHD filter and the multisensor CPHD filter. In our work, the general CPHD filter, which allows the clutter process and target measurement process to be arbitrary and performs better than the general PHD filter, is proposed. The proposed general CPHD filter is the general form of all kinds of CPHD filters, as well as all kinds of PHD filters and Bernoulli filters. In the simulation, the multiple-detection model of the over-the-horizon radar is used to demonstrate the tracking performance. The simulation results show that the proposed filter improves the accuracy of the state estimates and reduces the variance of the estimated number of targets.

Trajectory PHD and CPHD Filters With Unknown Detection Profile

Compared to the probability hypothesis density (PHD) and cardinalized PHD (CPHD) filters, the trajectory PHD (TPHD) and trajectory CPHD (TCPHD) filters are for sets of trajectories, and thus are able to produce trajectory estimates with better performance. In this paper, we develop the TPHD and TCPHD filters which can adaptively learn the history of the unknown target detection probability, and therefore they can perform more robustly in scenarios where targets are with unknown and time-varying detection probabilities. These filters are referred to as the unknown TPHD (U-TPHD) and unknown TCPHD (U-TCPHD) filters. By minimizing the Kullback-Leibler divergence (KLD), the U-TPHD and U-TCPHD filters can obtain, respectively, the best Poisson and independent identically distributed (IID) density approximations over a set of augmented trajectories. For computational efficiency, we also propose the U-TPHD and U-TCPHD filters that only consider the unknown detection profile at the current time. Specifically, the Beta-Gaussian mixture method is adopted for the implementations of proposed filters, which are referred to as the BG-U-TPHD and BG-U-TCPHD filters. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L$</tex-math></inline-formula> -scan approximations of these filters with much lower computational burden are also presented. Finally, various simulation results demonstrate that the BG-U-TPHD and BG-U-TCPHD filters can achieve robust tracking performance to adapt to unknown detection profile. Besides, it also shows that usually a small value of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L$</tex-math></inline-formula> in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L$</tex-math></inline-formula> -scan approximation has similar performance than with a large value at a fraction of the computational cost.

Robust CPHD Fusion for Distributed Multitarget Tracking Using Asynchronous Sensors

This paper studies the multitarget tracking problem based on an asynchronous network of sensors with different sampling rates, where each sensor runs a cardinalized probability hypothesis density (CPHD) filter. To fuse the filter estimates obtained at different sensors conditioned on asynchronous measurements, an arithmetic averaging approach is recursively carried out in a timely manner according to the network-wide sampling time sequence. The intersensor communication is conducted by a so-called partial flooding scheme, in which either cardinality distributions or intensity functions pertinent to local posteriors are disseminated among sensors. The fused results may not feedback to the filter, which will avoid communication delay to the local filters cased by intersensor fusion at the expense of reduced information gain. Furthermore, an extension of the proposed multi-sensor CPHD filter based on the bootstrap filtering algorithm is given to accommodate unknown clutter rate and detection profile. Numerical simulations are performed to test the proposed approaches.

Estimations of time-varying birth cardinality distribution and birth intensity in Gaussian mixture CPHD filter for multi-target tracking

Cardinalized probability hypothesis density (CPHD) filter is a promising algorithm for multi-target tracking. However, the performance of standard CPHD is degraded when the target birth process is unknown and dynamically changing. To remove this limitation, a discrete kernel estimator in conjunction with exponential weighted moving average scheme is introduced in this study to estimate the time-varying target birth cardinality distribution (i.e., probability distribution on number of newborn targets appearing during one sampling time) at each processing step. The target birth intensity is then updated according to the resulting estimated birth cardinality distribution. The estimated birth intensity and cardinality distribution can be employed by a tracker based on Gaussian mixture CPHD (GMCPHD) to modulate its filtering strength for target tracking. The performance of the proposed framework is demonstrated by both numerical simulations and experiment on a video surveillance dataset. Results show that the proposed algorithm can reduce the delay of standard GMCPHD filter's response to the changes in the number of targets, and thus improve the accuracy of cardinality estimates. Satisfactory estimation of cardinality can be obtained even when there is a fast change in the rate of birth.

Distributed multi-view multi-target tracking based on CPHD filtering

This paper addresses distributed multi-target tracking over a network of sensors having different fields-of-view (FoVs). Specifically, a cardinalized probability hypothesis density (CPHD) filter is run at each sensor node. Due to the fact that each sensor node has limited FoV, the standard fusion methods need to be suitably modified. In fact, the monitored area of multiple sensor nodes consists of several parts that are either exclusive of a single node, i.e. or common to multiple (at least two) nodes. In this setting, the crucial issue is how to account for these different information sets in the fusion rule. The problem is particularly challenging when the knowledge of the FoVs is unreliable, for example because of the presence of unknown occlusions, and when local sensors can have valuable information also on targets located outside the current nominal FoV, for example thanks to the diffusion of information in the network or because the FoVs are time-varying. In this context, we propose a distributed CPHD filter based on the idea of decomposing the posterior densities to be fused so as to account for the different information sets. A new decomposition method is proposed that does not rely on the nominal FoVs and, instead, uses clustering to decompose the posterior intensity function into multiple sub-intensities, and reconstructing the corresponding cardinality distribution via multi-Bernoulli approximation. Then, fusion is performed in parallel according to either the geometric average or arithmetic average rule. Simulation experiments are provided to demonstrate the effectiveness of the proposed approach.

Bearings-only multitarget tracking based onRao-Blackwellized particle CPHD filter

Following Mahler’s framework forinformation fusion, this paper develops a implementationof cardinalized probability hypothesis density (CPHD)filter for bearings-only multitarget tracking.Rao-Blackwellized method is introduced in the CPHDfiltering framework for mixed linear/nonlinear state spacemodels. The sequential Monte Carlo (SMC) method is usedto predict and estimate the nonlinear state of targets.Kalman filter (KF) is adopted to estimate the linear stateswith the information embedded in the estimated nonlinearstates. The multitarget state estimates are extracted byutilizing the kernel density estimation (KDE) theory andmean-shift algorithm to enhance tracking performance.Moreover, the computational load of the filter is analyzedby introducing equivalent flop measure. Finally, theperformance of the proposed Rao-Blackwellized particleCPHD filter is evaluated through a challengingbearings-only multitarget tracking simulation experiment.

Frequency tracking and mitigation method based on CPHD filter and adaptive multiple linear Kalman notch filter for multiple GNSS interference

In this paper, a frequency tracking method based on a cardinalized probability hypothesis density (CPHD) filter with cardinality compensation and fuzzy-c means (FCM) clustering is proposed to precisely estimate multiple interference frequencies in the received Global Navigation Satellite System (GNSS) signal. In addition, an adaptive multiple linear Kalman notch filter is applied in order to mitigate multiple GNSS interference signals by using tracking results from the proposed CPHD filter. The cardinality refers to the number of interference frequencies, and estimation accuracy on the cardinality has an effect on the tracking and mitigation performance of the filter. For that reason, the cardinality compensation process and FCM clustering are added in the CPHD filter. The proposed tracking and mitigation method is evaluated by theoretical analysis including simulations. It is confirmed that the proposed algorithm has better tracking and mitigation performance compared with the conventional algorithms.

Innovative unscented transform–based particle cardinalized probability hypothesis density filter for multi-target tracking

Multi-target tracking is widely applied in video surveillance systems. As we know, although the standard particle cardinalized probability hypothesis density filter can estimate state of targets, it is difficult to define the proposal distribution function in prediction stage. Since the robust particles cannot be effectively drawn, the actual tracking accuracy should be enhanced. In this paper, an innovative unscented transform–based particle cardinalized probability hypothesis density filter is derived. Considering the different state spaces, we use the auxiliary particle method and then draw robust particles from the modified distributions in order to estimate the position of targets. Simultaneously, we present the recursion of the optimized Kalman gain to improve the general unscented transform for the velocity estimates. Using the track label, we further integrate them in the framework of the jump Markov model. The simulation results show that the proposed filter has advances in the multi-target tracking scenes. Moreover, the experiments indicate that the filter can track mobile targets with satisfactory results.

Cardinality compensation method based on information-weighted consensus filter using data clustering for multi-target tracking

In this paper, a cardinality compensation method based on Information-weighted Consensus Filter (ICF) using data clustering is proposed in order to accurately estimate the cardinality of the Cardinalized Probability Hypothesis Density (CPHD) filter. Although the joint propagation of the intensity and the cardinality distribution in the CPHD filter process allows for more reliable estimation of the cardinality (target number) than the PHD filter, tracking loss may occur when noise and clutter are high in the measurements in a practical situation. For that reason, the cardinality compensation process is included in the CPHD filter, which is based on information fusion step using estimated cardinality obtained from the CPHD filter and measured cardinality obtained through data clustering. Here, the ICF is used for information fusion. To verify the performance of the proposed method, simulations were carried out and it was confirmed that the tracking performance of the multi-target was improved because the cardinality was estimated more accurately as compared to the existing techniques.

Distributed Multitarget Tracking Based on Diffusion Strategies Over Sensor Networks

We consider the distributed multitarget tracking over sensor networks, where each node only communicates with its neighbors. We develop a diffusion-based distributed multisensor multitarget tracking algorithm. The state update of the diffusion-based distributed algorithm is mainly composed of two phases: an adaptation phase and a combination phase. During the adaptation phase, each node updates its local estimate by using all its neighbors' measurements. It is achieved based on a multi-sensor cardinalized probability hypothesis density filter. During the combination phase, each node fuses all its neighbors' local estimates. It is achieved based on a generalized version of covariance intersection technique. Compared to the consensus-based distributed algorithm, the proposed algorithm has two advantages. First, it can provide more accurate and robust tracking results, especially when the detection probability that the sensors detect the targets is low. Second, it has lower communication load because the consensus iterations are not required. Numerical results are provided to illustrate the performance of the proposed algorithm.

A CPHD Approximation Based on a Discrete-Gamma Cardinality Model

The cardinalized probability hypothesis density (CPHD) filter has become one of the most acclaimed algorithms for multi-target Bayesian filtering due to its ability to accurately estimate the number of objects and the object states in tracking scenarios affected by clutter. The CPHD filter generalizes the probabilistic hypothesis density filter by jointly propagating the first-order multi-target moment (intensity function) along with the entire probability distribution on the number of targets (cardinality distribution). In general, the CPHD recursion is computationally intractable, however successful approximations have been devised with reported computational complexity dominated by $\mathcal{O}(m^{3})$ operations per filtering iteration, where $m$ is the number of measurements. Room for improvement was originally acknowledged by Mahler, who conceived the idea of approximating the cardinality distribution by two-parameter distributions. In this paper, we further explore this idea to provide an efficient approximation of the CPHD filter where the cardinality distribution is modeled as a discretized Gamma distribution. Experiments show that the resulting filter is less computationally complex than the standard implementation of the CPHD filter but shows similar cardinality accuracy and variance.

PHD and CPHD Filtering With Unknown Detection Probability

A priori knowledge of target detection probability is of critical importance in the Gaussian mixture probability hypothesis density (PHD) and cardinalized PHD (CPHD) filters. In addition, these two filters require that the process noise and measurement noise of the state propagated in the recursion be Gaussian. These limitations may restrict the two filters application in real problems. To accommodate unknown target detection probability and nonnegative non-Gaussian parameters, this paper proposes a new implementation based on inverse gamma Gaussian mixtures, introducing a location independent feature whose posterior probability density and likelihood function are nonnegative non-Gaussian inverse gamma and gamma functions to determine detection probability incorporated into the recursions. The derivation of the merging inverse gamma components is also presented to prevent the unbounded increase of mixture components by minimizing the Kullback–Leibler divergence. First, a real heavy-clutter scenario is used to validate the effectiveness of the proposed filters in track initiation and target tracking without known detection probability. Then, simulations are presented to demonstrate that the proposed CPHD and PHD filters can achieve multitarget tracking performance similar to the standard counterparts with known target detection probability, and that they outperform the standard counterparts in scenarios with unknown and dynamically changing detection probability. The robustness of the proposed filters is tested in both real and simulation scenarios. It is also shown that the analytical and empirical computational complexities of the proposed filters are similar to those of their standard counterparts.

CPHD Filter Birth Modeling Using the Probabilistic Admissible Region

Sparse observations make single-point track initialization difficult in some multitarget tracking scenarios, including space-object tracking. A probabilistic admissible region approach combines physics- and scenario-based constraints in unobservable directions to reduce ambiguity in the initial state and allows for a birth model consistent with the derivation assumptions in the cardinalized probability hypothesis density filter. The proposed filter enables tracking of simulated space objects via optical or radar observations and establishes custody of newly observed targets when given sparse observations.

A Covariance-Based Superpositional CPHD Filter for Multisource DOA Tracking

Direction of arrival (DOA) estimation of multiple sources using an array of sensors is a well-known problem in signal processing. Most DOA estimation methods use second-order measurements in the form of covariance matrices obtained from consecutive snapshots of the array elements’ raw data. In this paper, we pose the covariance-based DOA estimation problem in the random finite set framework of multitarget tracking using a superpositional model. The superpositional model allows for the measurements to be directly incorporated into a track-before-detect approximate cardinalized probability hypothesis density filter with a likelihood distributed as a complex Wishart random matrix that can perform DOA tracking with unknown time-varying number of sources. Complex Wishart and inverse-Wishart conjugacy is employed to derive the filter's update equations. The proposed filter is implemented using an auxiliary particle filter and simulation results showcase its improved performance in challenging scenarios of low (negative) signal-to-noise ratio and small number of snapshots.

Intrahour Cloud Tracking Based on Probability Hypothesis Density Filtering

Swift variations in the cloud cover may cause significant power output fluctuations in solar power systems, jeopardizing power quality. Forecasting the power output with a very short horizon below 30 s can be seen as a cloud cover forecasting problem solved by processing images from a sky camera. After a preprocessing stage that identifies clouds with a criterion based on the red-green blue values of each pixel, a probability hypothesis density filter or a more advanced cardinalized probability hypothesis density filter can be used to an unknown and varying number of clouds. The time when cumulus clouds will shade the sun can be forecasted with an absolute precision 6 s ahead and with an acceptable accuracy 27 s ahead. It has been found that both filters are equally well suited for a real-time online nowcasting application. They also have the potential to deal with noise and the swift dynamics and high variability of clouds.

Generalized Labeled Multi-Bernoulli Extended Target Tracking Based on Gaussian Process Regression

For the problems that Gamma Gaussian Inverse Wishart Cardinalized Probability Hypothesis Density (GGIW-CPHD) filter cannot accurately estimate the extended target shape and has a bad tracking performance under the condition of low SNR, a new generalized labeled multi-Bernoulli algorithm based on Gaussian process regression is proposed. The algorithm adopts the star convex to model the extended target, and realizes the online learning of the Gaussian process by constructing the state space model to complete the estimation of the extended target shape. At the same time, in the low SNR environment, the target motion state is tracked by the good tracking performance of the generalized label Bernoulli filter. Simulation results show that for any target with unknown shape, the proposed algorithm can well offer its extended shape and in the low SNR environment it can greatly improve the accuracy and stability of target tracking.

Labeled Random Finite Sets With Moment Approximation

The probability hypothesis density (PHD) filter was proposed as a practical approximation of the multitarget Bayes filter. The cardinalized PHD (CPHD) filter improves on the PHD filter by propagating cardinality distribution. However, both the PHD and CPHD filters have limitations in dealing with missed detections, extracting target state in their particle implementations, and maintaining track continuity. In this paper, based on the labeled random finite set theory, $\tau$ -labeled PHD and CPHD ( $\tau$ -LPHD/LCPHD) filtering approaches, in which the PHD components are separated and uniquely identified by a tag $\tau$ , are proposed. Based on the information from the $\tau$ -LPHD/LCPHD filtering processes, new $\delta$ -generalized labeled PHD and CPHD filtering approaches are proposed to address the issues in the $\tau$ -LPHD/LCPHD filters, which are similar to the issues in the standard PHD/CPHD filters. The two proposed methods interact with each other and inherit the advantages of the $\delta$ -generalized labeled multi-Bernoulli filter, with substantial savings in computation resources.