Cardinalized Probability Hypothesis Density Research Articles

Microbubble tracking based on partial smoothing-based adaptive generalized labelled Multi-Bernoulli filter for super-resolution imaging

Super-resolution ultrasound (SRUS) can image the vasculature at microscopic resolution according to microbubble (MB) localization, with velocity vector maps obtained based on MB tracking information. High MB concentrations can reduce the acquisition time of SRUS imaging, however adjacent and intersecting vessels are difficult to distinguish, thus decreasing resolution. Low acquisition frame rates affect the precision of flow velocity estimation. This study proposes a partial smoothing-based adaptive generalized labeled multi-Bernoulli filter (SAGLMB) to precisely track the MB motion at different flow velocities. SAGLMB employs a generalized labelled multi-Bernoulli filter (GLMB) for MB trajectory allocation to separate adjacent and intersecting vessels. Furthermore, the nonlinear motion of MB was predicted by an unscented Kalman filter, and a cardinalized probability hypothesis density filter was applied to suppress clutter interference. Finally, the trajectories were smoothed by unscented Rauch-Tung-Striebel to improve the resolution of the SRUS image. The simulation results demonstrate that SAGLMB outperforms the conventional bipartite graph-based tracking at high MB concentrations, achieving at least an 8.55 % improvement in the correctly paired precision, with 3 times increase in the structural similarity index measure. Moreover, SAGLMB can obtain more precise flow velocity estimations with a 4 times improvement than the conventional method. The SRUS results of rabbit kidney show that the proposed method significantly improves resolution of adjacent and intersecting vessels at higher MB concentrations and maintains this performance as the acquisition frame rate decreases. Furthermore, the rat brain microvascular network was reconstructed with 9.21 μm (λ/11.1) resolution. Therefore, SAGLMB can achieve robust SRUS imaging at high concentrations and low acquisition frame rates.

A Robust TCPHD Filter for Multi-Sensor Multitarget Tracking Based on a Gaussian–Student’s t-Mixture Model

To realize multitarget trajectory tracking under non-Gaussian heavy-tailed noise, we propose a Gaussian–Student t-mixture distribution-based trajectory cardinality probability hypothesis density filter (GSTM-TCPHD). We introduce the multi-sensor GSTM-TCPHD (MS-GSTM-TCPHD) filter to enhance tracking performance. Conventional cardinality probability hypothesis density (CPHD) filters typically assume Gaussian noise and struggle to accurately establish target trajectories when faced with heavy-tailed non-Gaussian distributions. Heavy-tailed noise leads to significant estimation errors and filter dispersion. Moreover, the exact trajectory of the target is crucial for tracking and prediction. Our proposed GSTM-TCPHD filter utilizes the GSTM distribution to model heavy-tailed noise, reducing modeling errors and generating a set of potential target trajectories. Since single sensors have a limited field of view and limited measurement information, we extend the filter to a multi-sensor scenario. To tackle the issue of data explosion from multiple sensors, we employed a greedy approximation method to assess measurements and introduced the MS-GSTM-TCPHD filter. The simulation results demonstrate that our proposed filter outperforms the CPHD/TCPHD filter and Student’s t-based TCPHD filter in terms of accurately estimating the trajectories of multiple targets during tracking while also achieving improved accuracy and shorter processing time.

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.

An Efficient Implementation Method for Distributed Fusion in Sensor Networks Based on CPHD Filters.

A highly efficient implementation method for distributed fusion in sensor networks based on CPHD filters is proposed to address the issues of unknown cross-covariance fusion estimation and long fusion times in multi-sensor distributed fusion. This method can effectively and efficiently fuse multi-node information in multi-target tracking applications. Discrete gamma cardinalized probability hypothesis density (DG-CPHD) can effectively reduce the computational burden while ensuring computational accuracy similar to that of CPHD filters. Parallel inverse covariance intersection (PICI) can effectively avoid solving high-dimensional weight coefficient convex optimization problems, reduce the computational burden, and efficiently implement filtering fusion strategies. The effectiveness of the algorithm is demonstrated through simulation results, which indicate that PICI-GM-DG-CPHD can substantially reduce the computational time compared to other algorithms and is more suitable for distributed sensor fusion.

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.

The trajectory motion model based TPHD and TCPHD filters for maneuvering targets

In this article, we present the TMM-TPHD and TMM-TCPHD filters, which are the alternative trajectory probability hypothesis density (TPHD) and the alternative trajectory cardinality probability hypothesis density (TCPHD) filters for tracking maneuvering targets. To describe the trajectory maneuver, we define the motion model history sequence of a trajectory as the trajectory motion model (TMM) variable. By incorporating the TMM, we augment the general trajectory model to accommodate the maneuver. Based on this augmented model, we propose the TMM extensions of the standard TPHD and TCPHD filters, namely TMM-TPHD and TMM-TCPHD filters. These filters can directly estimate the number, trajectory state, and TMM of maneuvering targets. In addition to the derivation of the filtering formulas, we also develop linear Gaussian mixture (LGM) implementations for TMM-TPHD and TMM-TCPHD filters. We verify the tracking performance and parameter stability of the proposed filters through tracking simulations involving multiple maneuvering targets.

Vardiational Bayesian Hybrid Multi-Bernoulli and CPHD Filters for Superpositional Sensors

This paper addresses the problem of multi-target tracking with superpositional sensors, while the covariance matrices of measurement noise are not known. The proposed method is based on the hybrid multi-Bernoulli cardinalized probability hypothesis density (HMB-CPHD) filter, which has been developed for superpositional sensors-based multi-target tracking with known measurement noises. Specifically, we firstly propose the Gaussian mixture (GM) implementation of the HMB-CPHD filter, and then the covariance matrices of measurement noises are augmented into the target state vector, resulting in the Gaussian and inverse Wishart mixture (GIWM) representation of the augmented state. Then the variational Bayesian (VB) method is exploited to approximate the posterior distribution so that it maintains the same form as the prior distribution. A remarkable feature of the proposed method is that it can jointly perform multi-target tracking and measurement noise covariance estimation. The performance of the proposed algorithm is demonstrated via simulations.

Gaussian Mixture Cardinalized Probability Hypothesis Density(GM-CPHD): A Distributed Filter Based on the Intersection of Parallel Inverse Covariances

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.

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.

The trajectory CPHD filter for spawning targets

This paper proposes a new trajectory cardinality probability hypothesis density (TCPHD) filter to track multiple spawning targets, referred to as the S-TCPHD filter. Firstly, we build the spawning events of single trajectory as a zero-inflated Poisson process and present corresponding trajectory set dynamic model. Then, the prediction equations of the S-TCPHD filter is derived based on the dynamic model, while the update step is the same as the TCPHD. In addition, we also develop the linear Gaussian mixture (LGM) implementations of the prediction step. The resulting S-TCPHD filter can estimate the trajectory information of targets, thereby being versatile and suitable for spawning target applications. The trajectory probability hypothesis density (TPHD) filter for spawning targets, namely S-TPHD, can be directly obtained from the S-TCPHD filter. Finally, the simulation results verify the tracking performance of the proposed S-TCPHD/S-TPHD filters in a multiple spawning targets tracking scenario.

Joint DOA tracking and spectral signature estimation of wideband sources using a multi-spectral TBD-CPHD filter

In this paper, a novel method for direction of arrival (DOA) tracking of multiple unknown time-varying number of wideband emitters is presented using a random finite set (RFS) framework. This method combines multiple narrowband covariance-based measurements to form a wideband cardinalized probability hypothesis density (CPHD) filter using the parallel combination (PC) Lemma generally used in multisensor fusion. By doing this, each narrowband frequency bin acts as a separate virtual spectral sensor. The covariance-based measurements, represented by complex Wishart random matrices, also allow the method to have a track-before-detect (TBD) property, which has shown to be advantageous in very low signal-to-noise ratios (SNR). Furthermore, the proposed method can simultaneously monitor the spectrum of the environment and extract the spectral characteristics of the active sources, i.e. can work as some sort of cognitive radio (CR). The suggested algorithm is implemented using an auxiliary particle filter. The simulation results and the comparison with the known state-of-the-art approaches confirm the superiority of the proposed method in negative SNR scenarios and low number of snapshots.

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.

Kernel-based learning of birth process from evolving spatiotemporal RFS data stream in SMC[sbnd]CPHD filter for multi-target tracking

This paper presents a kernel-based framework to estimate the birth process in the sequential Monte Carlo cardinalized probability hypothesis density (SMCCPHD) filter from the evolving spatiotemporal birth random finite set data stream that can enhance the performance of the SMCCPHD based trackers when tracking targets with no a priori information on where and when targets can appear. The time-varying spatial distribution of birth events is modelled as a map with adaptive grid points and their corresponding density values. While the spatial distribution of birth density is estimated from the elements of the birth random finite sets (RFSs), the time-varying birth cardinality distribution is estimated from the cardinality of the birth RFSs. The corresponding estimations are eventually used by a SMCCPHD based tracker to adjust its filtering strength spatially and temporally for target tracking. Experimental results on both the synthetic and real video surveillance datasets show that the proposed algorithm can improve the tracking accuracy by more precisely eliminating clutters and more effectively capturing newborn 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.

SMC-CPHD Filter with Adaptive Survival Probability for Multiple Frequency Tracking

We propose a sequential Monte Carlo-based cardinalized probability hypothesis density (SMC-CPHD) filter with adaptive survival probability for multiple frequency tracking to enhance the tracking performance. The survival probability of the particles in the filter is adjusted using the pre-designed exponential function related to the distribution of the estimated particle points. In order to ensure whether the proposed survival probability affects the stability of the filter, the error bounds in the prediction process are analyzed. Moreover, an inverse covariance intersection-based compensation method is added to enhance cardinality tracking performance by integrating two types of cardinality information from the CPHD filter and data clustering process. To evaluate the proposed method’s performance, MATLAB-based simulations are performed. As a result, the tracking performance of the multiple frequencies has been confirmed, and the accuracy of cardinality estimates are improved compared to the existing filters.

Asynchronous Non–uniform Distributed Multi–target Tracking Filter Based on Asymmetric Alpha–divergence Consensus

With more and more extensive application of target tracking, distributed multi–target tracking (DMT) becomes an important research direction. However, the synchronization characteristics of some sensor networks can not be guaranteed due to communication delay, non–uniform sampling and so on. Therefore, a DMT algorithm for asynchronous non–uniform multi–sensor networks is studied in this paper. Firstly, by using the cardinalized probability hypothesis density (CPHD) based on continuous–discrete multi–target dynamic (CD) model (CD-CPHD), a CD–CPHD algorithm with multi–step birth process and time–triggered structure (TCD–CPHD) is proposed; Secondly, an asymmetric alpha–divergence (AAD) based adaptive computing structure of time–trigger index is designed to reduce the communication rate of TCD–CPHD; Thirdly, the fusion rule of AAD consensus is derived to construct the fusion process of the proposed algorithm; Finally, by combining the above structures and the fusion rule, the implementation process of the proposed algorithm is given. Theoretical analysis and exhaustive experimental analysis show the effectiveness of the proposed algorithm.

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.

Distributed GGIW-CPHD-Based Extended Target Tracking Over a Sensor Network

Multiple extended target tracking (METT) is a common and challenging problem. Various solutions for METT have been proposed, however, most of them focus on the single-sensor or centralized multi-sensor scenarios. In this letter, we explore the multi-sensor METT problem in a distributed fusion framework. Specifically, there are two stages in the implementation process: 1) to perform a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Gamma Gaussian Inverse Wishart Cardinalized Probability Hypothesis Density</i> (GGIW-CPHD) filter for each sensor node, and 2) to perform a fusion by resorting to the so-called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Generalized Covariance Intersection</i> (GCI) fusion rule. In the fusion stage, we derive an approximate GGIW mixture form of the fused spatial density. Lastly, simulation experiments via a consensus sensor network are provided to verify the effectiveness of the proposed approach.

Robust multi-sensor generalized labeled multi-Bernoulli filter

This paper proposes an efficient and robust algorithm to estimate target trajectories with unknown target detection profiles and clutter rates using measurements from multiple sensors. In particular, we propose to combine the multi-sensor Generalized Labeled Multi-Bernoulli (MS-GLMB) filter to estimate target trajectories and robust Cardinalized Probability Hypothesis Density (CPHD) filters to estimate the clutter rates. The target detection probability is augmented to the filtering state space for joint estimation. Experimental results show that the proposed robust filter exhibits near-optimal performance in the sense that it is comparable to the optimal MS-GLMB operating with true clutter rate and detection probability. More importantly, it outperforms other studied filters when the detection profile and clutter rate are unknown and time-variant. This is attributed to the ability of the robust filter to learn the background parameters on-the-fly.

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.