Posterior Cramer-Rao Lower Bound Research Articles

Seamless Tracking of Apparent Point and Extended Targets Using Gaussian Process PMHT

In practical target tracking scenarios, targets of different sizes (or extents) may be near or far away from the sensor, which may result in targets appearing as point sources or as extended targets spanning one or more resolution cells, respectively, depending on distance and sensor resolution. In this paper, a new Gaussian Process (GP) measurement model is proposed to explicitly describe the observation about each basis point of GP by an individual dynamic Poisson measurement rate. By employing this model, a novel algorithm to track multiple point targets and extended targets, simultaneously and seamlessly, in the presence of clutter and missed detections is proposed within the Probabilistic Multi-Hypothesis Tracker (PMHT) framework. The proposed algorithm can adapt to spatio-temporally varying target sizes or extents of extended targets and temporally varying target cardinality. In addition, the posterior Cramer-Rao lower bound (PCRLB) for extended targets, which quantifies the accuracies of estimates of multiple extended target states in scenarios with clutter, is derived. Simulations with a scenario consisting of multiple extended targets and point targets are used to verify the effectiveness of the proposed algorithm and to compare its performance with the extended target PCRLB and with those of existing extended target tracking algorithms.

Analysis of Byzantine Attacks for Target Tracking in Wireless Sensor Networks.

Herein, the problem of target tracking in wireless sensor networks (WSNs) is investigated in the presence of Byzantine attacks. More specifically, we analyze the impact of Byzantine attacks on the performance of a tracking system. First, under the condition of jointly estimating the target state and the attack parameters, the posterior Cramer–Rao lower bound (PCRLB) is calculated. Then, from the perspective of attackers, we define the optimal Byzantine attack and theoretically find a way to achieve such an attack with minimal cost. When the attacked nodes are correctly identified by the fusion center (FC), we further define the suboptimal Byzantine attack and also find a way to realize such an attack. Finally, in order to alleviate the negative impact of attackers on the system performance, a modified sampling importance resampling (SIR) filter is proposed. Simulation results show that the tracking results of the modified SIR filter can be close to the true trajectory of the moving target. In addition, when the quantization level increases, both the security performance and the estimation performance of the tracking system are improved.

LaIF: A Lane-Level Self-Positioning Scheme for Vehicles in GNSS-Denied Environments

Vehicle self-positioning is of significant importance for intelligent transportation applications. However, accurate positioning (e.g., with lane-level accuracy) is very difficult to obtain due to the lack of measurements with high confidence, especially in an environment without full access to a global navigation satellite system (GNSS). In this paper, a novel information fusion algorithm based on a particle filter is proposed to achieve lane-level tracking accuracy under a GNSS-denied environment. We consider the use of both coarse-scale and fine-scale signal measurements for positioning. Time-of-arrival measurements using the radio frequency signals from known transmitters or roadside units, and acceleration or gyroscope measurements from an inertial measurement unit (IMU) allow us to form a coarse estimate of the vehicle position using an extended Kalman filter. Subsequently, fine-scale measurements, including lane-change detection, radar ranging from the known obstacles (e.g., guardrails), and information from a high-resolution digital map, are incorporated to refine the position estimates. A probabilistic model is introduced to characterize the lane changing behaviors, and a multi-hypothesis model is formulated for the radar range measurements to robustly weigh the particles and refine the tracking results. Moreover, a decision fusion mechanism is proposed to achieve a higher reliability in the lane-change detection as compared to each individual detector using IMU and visual (if available) information. The posterior Cramer–Rao lower bound is also derived to provide a theoretical performance guideline. The performance of the proposed tracking framework is verified by simulations and real measured IMU data in a four-lane highway.

GP-PDA Filter for Extended Target Tracking With Measurement Origin Uncertainty

Extended target tracking (ETT) is an issue in high-resolution radar surveillance, ship tracking, and video tracking. Most of the previous works focus on tracking an ellipsoidal extended target without measurement origin uncertainty (missed detections and clutter). In this paper, a new estimator called the Gaussian process probabilistic data association (GP-PDA) filter is proposed to track an irregularly shaped extended target with measurement origin uncertainty. First, a generalized measurement model for ETT using the Gaussian process (GP) is presented. Both the interior scattering points and the external clutter are considered in this model. Second, a GP-based gating technique is constructed to select validated measurements to feed the filter. Third, the GP-PDA filter is proposed to simultaneously estimate the kinematic state and the contour state of the extended target with measurement origin uncertainty. It is proven that the GP-PDA is a generalized version of the classic PDA, and the latter is a special case of the former in the point target tracking applications. Finally, the GP-based posterior Cramer–Rao lower bound (PCRLB) is derived to evaluate the performance of the ETT with measurement origin uncertainty. Two cases of the PCRLB are discussed, with the number of scattering points being known and unknown. Simulation results verify the effectiveness of the proposed method.

Globally Valid Posterior Cramér–Rao Bound for Three-Dimensional Bearings-Only Filtering

This paper addresses the problem of three-dimensional target motion analysis from bearing measurements. The established description of uncertain directional data involves azimuth and elevation angles with additive Gaussian noise. Investigation of this widely accepted representation in terms of the corresponding Fisher information matrices reveals that the underlying probability densities are only valid for small absolute elevation angles and precise measurements. Employing spherical statistics, an alternative representation is proposed that is sensible for the whole state space and arbitrarily imprecise bearing sensors. Based on this, a globally valid posterior Cramer–Rao lower bound is derived.

Optimal bit allocation for maneuvering target tracking in UWSNs with additive and multiplicative noise

Due to the limited energy and bandwidth in underwater wireless sensor networks(UWSNs), the original measurements should be quantized before transmitted to the fusion center. In this paper, the optimal bit allocation is utilized to improve the tracking accuracy on the premise of transmitting the same number of quantized bits of data. The relationship between the quantization level of sensor nodes and posterior Cramer–Rao lower bounds (PCRLB) is derived and taken as the performance bound for tracking accuracy. Then the problem of optimal bit allocation is converted into an optimization problem. To make computation-efficiency, allocating the maximum bits for each candidate sensors and then the generalized Breiman, Friedman, Olshen, and Stone (GBFOS) algorithm is adopted to delete one bit at a time until the total number of bits is satisfied. In addition, the genetic algorithm for bit allocation (GABA) is proposed in this paper to solve the optimization problem when the transmission bits and the number of candidate sensors are large.The simulation results illustrate the performance of the proposed scheme in improving the tracking accuracy on the condition of limited communication bandwidth. Compared to GBFOS, GABA proposed in this paper can satisfy the real-time target tracking requirements and ensure tracking performance.

Space-time allocation for transmit beams in collocated MIMO radar

In this paper, we propose a space-time allocation strategy for collocated multiple-input multiple-output (MIMO) radar to determine the beam directions to multiple targets, to be tracked. The transmit beams are allocated to different targets with distinct time units in one illumination. The basis of our strategy is to achieve the optimal probing through the feedback information in the tracking recursion cycle, with the objective of improving the overall tracking accuracy of multiple targets. We derive the posterior Cramer-Rao lower bound (PCRLB), used as the optimization criterion, to quantify the target state estimate accuracy. By introducing the illumination probability in each time unit, the space-time allocation problem is converted to a convex one. Therefore, a sufficient solution is provided, which aims at the real-time demand of radar system. Simulation results demonstrate the superiority of the proposed strategy to the uniform allocation, space-only allocation and time-only allocation methods.

Multi-vehicle tracking with microscopic traffic flow model-based particle filtering

This paper addresses the problem of tracking multiple vehicles on multi-lane roads with consideration for interactions among vehicles. Due to limited lane resources and traffic heterogeneity, vehicles have to interact with neighboring vehicles while moving along roads to improve their navigability and safety, resulting in highly dependent motions. However, multitarget tracking algorithms generally assume that targets move independently of one another. To address this limitation, using the microscopic traffic flow (MTF) model for modeling vehicle dynamics in the presence of interactions with surrounding traffic, an MTF-based tracking algorithm is proposed under the particle filter (PF) framework. The recursive maximum likelihood (RML) method is integrated into the PF to estimate unknown parameters in the MTF model. The posterior Cramer–Rao lower bound (PCRLB) is also derived for this problem. The performance of the proposed MTF-PF algorithm is compared with those of existing algorithms for multi-vehicle tracking on multi-lane roads. Numerical results show that the proposed algorithm requires less prior information while yielding more accurate and consistent tracks.

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

Abstract Multi-target tracking is an important problem in civilian and military applications. This paper investigates multi-target tracking in distributed sensor networks. Data association, which arises particularly in multi-object scenarios, can be tackled by various solutions. We consider sequential Monte Carlo implementations of the Probability Hypothesis Density (PHD) filter based on random finite sets. This approach circumvents the data association issue by jointly estimating all targets in the region of interest. To this end, we develop the Diffusion Particle PHD Filter (D-PPHDF) as well as a centralized version, called the Multi-Sensor Particle PHD Filter (MS-PPHDF). Their performance is evaluated in terms of the Optimal Subpattern Assignment (OSPA) metric, benchmarked against a distributed extension of the Posterior Cramer-Rao Lower Bound (PCRLB), and compared to the performance of an existing distributed PHD Particle Filter. Furthermore, the robustness of the proposed tracking algorithms against outliers and their performance with respect to different amounts of clutter is investigated.

Optimal Estimation and Fundamental Limits for Target Localization Using IMU/TOA Fusion Method

Localization is one of the most important topics of the cyber-physical system. In the last decades, much attention has been paid to the precise localization and performance evaluation in wireless sensor networks. The inertial measurement unit (IMU) and time-of-arrival (TOA) fusion is a state-of-the-art method to solve the accumulative error and drifting problem faced by the sole IMU positioning and navigation method. Many of the existing studies are based on optimization. However, they usually face problems of non-convexity of the objective function, falling into local optimum, and the requirements for the prior/posterior probability distribution of measured values. All these reasons limit its practical applications toward accurate target tracking. This paper presents a Chebyshev-center-based optimization method. Geometrically considering the real position of the target, it aims at improving the target tracking accuracy. Cramer-Rao lower bound (CRLB) and posterior CRLB for IMU/TOA fusion are derived to characterize both the spatial and temporal localization performance of the proposed fusion method. The simulation results show that the proposed fusion method in this paper has obvious spatial-temporal performance advantages in theory. Practical use cases are also conducted, and the experimental results show that the proposed method significantly decreases the drift errors and has a lower tracking error compared with the state of the art.

Posterior Cramer‐Rao lower bound for wireless sensor localisation networks

A general closed expression of posterior Cramer-Rao lower bound (PCRLB) for wireless sensor networks is derived and presented, based on Bayesian estimation and well suited to dynamic systems. The aim is to guide the localisation algorithm design and anchor layout fulfilling certain accuracy requirement. There is a growing need for wireless sensor systems to determine the position information indoors and in somewhere satellite positioning cannot provide service. PCRLB sets a fundamental lower limit to all kinds of localisation algorithms within consideration of dynamic state transition. This result shows that the theoretically best position solution can be obtained.

Receive-Beam Resource Allocation for Multiple Target Tracking With Distributed MIMO Radars

A distributed multiple-input multiple-output (MIMO) radar system is capable of tracking multiple targets under the “defocused transmit-focused receive” (DTFR) operating mode, in which each transmitter forms a completely defocused beam to illuminate the whole surveillance region and each receiver adopts a focused beam to attain a higher resolution. However, when operating in this mode, there exists a resource optimization problem on the allocation of receive-beams. To address this problem, a receive-beam resource allocation (RBRA) strategy is proposed in this paper. The key mechanism is to implement the optimal allocation between receive-beams and targets based on the feedback information in the tracking recursion cycle, with the objective of improving the worst tracking accuracy with multiple targets. Since the posterior Cramer–Rao lower bound (PCRLB) provides a lower bound on the accuracy of target state estimates, it is derived and adopted as an optimization criterion. It is shown that the optimal RBRA is a multidimensional nonconvex assignment problem that is NP-hard. We propose an efficient convex relaxation optimization to solve it. Numerical results demonstrate the superior performance of the proposed strategy in terms of the worst case tracking root mean-square errors.

Adaptive Sensor Selection for Nonlinear Tracking via Sparsity-Promoting Approaches

In this paper, we propose a few sparsity-promoting approaches for sensor selection in wireless sensor networks (WSNs) to reach the optimal tradeoff between the tracking performance and the sensing-communication cost. The proposed framework is valid for general nonlinear target tracking without explicit linearization and not restricted to any specific estimator. We formulate the sensor selection problem as the design of a sparse selection vector. The cardinality of the selection vector is added as a sparsity-promoting penalty term to the cost function where the conditional posterior Cramer–Rao lower bound is used as the criterion for sensor selection. To cope with large-scale WSNs, by combining iterative reweighted $\ell _1$ -norm minimization with the accelerated proximal gradient method and the alternating direction method of multipliers (ADMM), we develop two efficient sensor selection algorithms, respectively. We further develop a low-complexity distributed version of the ADMM where each sensor makes a local sensor selection decision. We test the proposed algorithms by simulations based on an extended Kalman filter using analog data and a particle filter using quantized data, respectively, which provide valuable insights and demonstrate the proposed algorithms’ performance.

Mutual-information based weighted fusion for target tracking in underwater wireless sensor networks

Underwater wireless sensor networks (UWSNs) can provide a promising solution to underwater target tracking. Due to limited energy and bandwidth resources, only a small number of nodes are selected to track a target at each interval. Because all measurements are fused together to provide information in a fusion center, fusion weights of all selected nodes may affect the performance of target tracking. As far as we know, almost all existing tracking schemes neglect this problem. We study a weighted fusion scheme for target tracking in UWSNs. First, because the mutual information (MI) between a node’s measurement and the target state can quantify target information provided by the node, it is calculated to determine proper fusion weights. Second, we design a novel multi-sensor weighted particle filter (MSWPF) using fusion weights determined by MI. Third, we present a local node selection scheme based on posterior Cramer-Rao lower bound (PCRLB) to improve tracking efficiency. Finally, simulation results are presented to verify the performance improvement of our scheme with proper fusion weights.

Geometrical kinematic modeling on human motion using method of multi-sensor fusion

Human motion sensing based on wearable sensors could be viewed as a multi-objects tracking issue of human body joints. Sensor drift errors and distortion are the main challenges of the tracking accuracy of human motion. Traditional filtering and fusion methods, such as Kalman filter, can to some extent reduce the instantaneous error but cannot avoid sensor drift fundamentally. Physical characteristics of human body should be considered and human motion models should be exhibited to describe human motions. The existing models such as skeleton model and cylinder model are either too simple or too complicated for practical applications. In this study, we put forward a geometrical kinematic characteristics based human motion model. The whole human body is viewed as an articulated skeleton and Denavit–Hartenberg convention is adopted to describe the forward kinematics structure. Theoretical analysis is conducted with the derivation of Posterior Cramer–Rao Lower Bound (PCRLB) in human movement scenes based on proposed model. Significant superiority is shown in simulation results. An experiment on human lower limb motions is carried out to verify the validity of the proposed human motion model in practice applications, from the angles of both capturing accuracy and energy consumption. The capturing accuracy has an obvious increase in the testing results, with acceptable energy consumption. It is far more efficient than traditional methods.

Fundamental Limits of TOA/DOA and Inertial Measurement Unit-Based Wireless Capsule Endoscopy Hybrid Localization

In this paper, performance analysis of hybrid localization based on radio-frequency (RF) and inertial measurement unit (IMU) measurements for a single wireless capsule endoscopy (WCE) traveling the gastrointestinal tract is studied. Specifically, the multiple body-mounted sensors are considered which are located on the front and back of a patient’s medical jacket and form the uniform rectangular arrays (URAs). With the aim of locating the WCE, two types of RF measurements, namely time-of-arrival (TOA) and direction-of-arrival (DOA), are estimated from the received signals at the URAs transmitted by the WCE, which are integrated with the IMU acceleration measurements via the standard extended Kalman filter. Here, a posterior Cramer–Rao Bound (PCRB) of the proposed TOA/DOA and IMU-based hybrid localization is derived as fundamental limits on squared position error, where the accuracies of TOA and DOA measurements are entailed by means of CRB to account for their dependency on the environmental parameters, while the accuracies of the IMU measurements are addressed with the acceleration measurement error standard deviation. Numerical results are provided, sustained by simulations which verify the millimeter accuracy of the TOA/DOA and IMU-based hybrid localization within the regulation of medical implant communication services and the exactness of the PCRB.

Design, Implementation, and Fundamental Limits of Image and RF Based Wireless Capsule Endoscopy Hybrid Localization

Precise localization of Wireless Capsule Endoscopy (WCE) inside the curly, long, and compact small intestine remains a challenging problem facing researchers for more than a decade. Conventional Radio Frequency (RF) localization techniques, commonly used in outdoor and indoor area, have demonstrated a few centimeters accuracy when applying to the inside of human body. In this paper, using 3D Posterior Cramer-Rao Lower Bound (PCRLB) as a framework for performance evaluation, we demonstrated that millimetric accuracy can be achieved using hybrid RF and image processing localization technique. This level of accuracy enables precise simultaneous localization and mapping of the WCE movement path inside the small intestine. Using the PCRLB framework, we provided in-depth analysis on hybrid localization performance regarding the effects of WCE movement estimation, the effects of system bandwidth as well as the effects of on-body sensor numbers and placements.

Posterior Cramer–Rao lower bounds for complicated multi-target tracking with labeled FISST based filters

As science develops, multi-target tracking technique advances towards dealing with complicated scenes, in which target number is time-varying and unknown, detection, measurement source and data association are uncertain. Among the new tracking methods, Mahler׳s Finite Set Statistics (FISST) based multi-target tracking technology naturally suits such complicated scenes. A performance metric with rigid theoretical explanation and clear physical connotation is the cornerstone for further improving multi-target tracking methods on precision and stability. The Posterior Cramer–Rao lower bounds (PCRLB) is widely used for assessing tracking performance. While, for the complicated multi-target tracking mentioned above, existing PCRLBs do not work well. Therefore, we derived multi-target PCRLB (MT-PCRLB) under random finite set frame as well as its iterative expression through analyzing lower bound of the FISST based filters’ performance in complicated multi-target tracking. The derived lower bound is compatible with current labeled FISST based filters. Based on multi-target tracks obtained by labeled FISST based filters and association relations between tracks and measurement set, recursive calculation of MT-PCRLB is realized. Simulation results demonstrate that the proposed methods behave in a manner consistent with our expectations.

The Effect of Sensor Modality on Posterior Cramer-Rao Bounds for Simultaneous Localisation and Mapping

This paper applies Posterior Cramer-Rao Bound theory to the SLAM problem to measure the information supplied by different sensor modalities over time. Range-only, bearing-only and full range-bearing sensors were considered, as well as the gain in information achieved by using multiple sensors in centralized co-operative SLAM. An efficient recursive formula was used to compute the bound for a set of simulated scenarios, and its validity verified by comparing the bound with the second-order error performance of Fast SLAM 2.0 and the EKF.

A Diffraction Measurement Model and Particle Filter Tracking Method for RSS-Based DFL

Device-free localization (DFL) based on received signal strength (RSS) measurements functions by measuring RSS variation due to the presence of the target. The accuracy of a certain localization method closely depends on the accuracy of the measurement model itself. Existing models have been found not accurate enough under certain circumstances as they cannot explain some phenomena observed in DFL practices. In light of this, we propose a new model to characterize the RSS variation, which invokes diffraction theory and regards the target as a cylinder instead of a point mass. It is observed that the proposed model agrees well with experimental measurements, particularly when the target crosses the link or is in the vicinity of the link. Since the proposed measurement model is highly nonlinear, a particle filter-based tracking method is used to generate the approximate Bayesian estimate of the target position. As a performance benchmark, we have also derived the posterior Cramer–Rao lower bound of DFL for a diffraction model. A field test has shown that the proposed diffraction model may improve the tracking accuracy at least by 45% in a single-target case and by 27% in a double-target case.