Kalman filtering is widely used for object tracking applications but often relies on predefined motion models, which limits its adaptability to nonlinear or uncertain trajectories. Formulating a Gaussian Process (GP) as a linear Gaussian state-space model provides a data-driven alternative for efficient sequential inference within a Kalman filtering framework. This paper extends the Integrated GP (iGP) model to introduce three novel variants: the Twice-Integrated GP (iiGP), the Dynamics-Informed Integrated GP (iDGP), and the Dynamics-Informed Twice-Integrated GP (iiDGP). The dynamics-informed models incorporate system dynamics alongside data-driven modelling, enabling more accurate tracking of motion under uncertain environments. The twice-integrated models enforce smoother motion patterns while maintaining the flexibility of GPs as driving noise. These models are implemented in the Stone Soup tracking framework. We discuss the software design, implementation challenges, and present evaluation results on synthetic and insect motion trajectory data. Results highlight the benefits of incorpo-rating dynamical information in GP-based tracking models, as well as the trade-offs between adaptability and robustness.

Existing trackers based on Poisson measurement process often struggle with efficiency and accuracy in large-scale tracking under heavy clutter. To overcome this, we introduce PiVoT, a scalable, robust multi-object tracker capable of efficiently detecting and tracking a large, varying number of objects, along with their shapes, existence probabilities, and measurement rates, even in heavy clutter. PiVoT employs a novel two-stage variational inference routine to achieve inference tractability and closed-form, parallelisable updates. Efficiency is further enhanced by early identification and removal of ineffective birth objects and designing highly simplified, much faster, yet equivalent variational updates. Additionally, PiVoT inherently offers efficient clutter-robust clustering, an innovation that can also enhance existing trackers that depend on supplementary clustering techniques. Experiments demonstrate PiVoT's clear accuracy and efficiency gains over existing methods, while also highlighting its ability to track a thousand closely spaced objects in under a second on a standard laptop without gating.

Lévy state-space models (SSMs) are a class of Bayesian models exhibiting heavy-tailed dynamics, particularly suited to systems with extreme values and abrupt changes. A broad family of Lévy SSMs that admit a linear-Gaussian conditional sub-model can be filtered using the Rao-Blackwellized particle filter (RBPF). In this work, we implement an efficient form of particle MCMC, taking advantage of end-to-end GPU acceleration and the special structure of the Lévy SSM, to permit inference in a practical time frame. We benchmark the performance of our implementation against a comparable CPU implementation, as well as existing comparable software packages, before showcasing our inference methods on a financial time series example, performing efficient joint inference on the SSM parameters alongside a novel Bayesian nonparametric estimate of the driving Lévy measure.

In recent years, state-space models for highly manoeuvrable objects have been proposed based on non-Gaussian, continuous time, jump-based Lévy processes, the so-called Lévy state-space model [1]–[4]. In these models, the standard Brownian motion driving process for continuous time processes is replaced with a heavy-tailed non-Gaussian alternative. This retains all the flexibility of its Gaussian counterpart in terms of possible dynamical model structures and operations with irregular time stamps or heterogeneous data sources. These models aim to operate in areas such as surveillance of irregularly moving drones or people, and tracking wildlife or biological data. Implementation is relatively straightforward since the Kalman filters of the Brownian motion case can be replaced in the nonGaussian case by mixtures of Kalman filters within a marginalised particle filtering framework [5]. While the Stone Soup tracking software environment includes both Kalman filtering and generic particle filtering, it does not currently allow the combination of these tasks within a marginalised particle filtering framework. We discuss the significant challenges involved in incorporating these models and algorithms into Stone Soup, and present initial simulation results for the new software.

This paper presents an adaptive approach to real-time multi-object localisation in addition to Siteswap inference, and performance evaluation metrics for juggling routines, employing a proposed bimodal machine learning-enhanced state-space model implementation. Considering the complex multi-modal characteristics exhibited by objects during performances, the paper introduces a bespoke Interacting Multiple Model (IMM) component for increased Siteswap beat detection accuracy and gravitational acceleration inference, and a scheme for causal Siteswap inference derived through machine learning-enhanced IMM mode outputs. The algorithm effectively models the transitory behaviour of the system, enabling rapid and smooth transitions between the two discrete tracking cases (airborne, and caught) and accurate Siteswap inference under a variety of camera and environmental conditions. The employment of beat tracking algorithms that exploit optimal compromises in time domain onset detection functions and Tempograms, enables effective error correction of Siteswap detections, in addition to providing performance analysis and visualisation utilities. Experimentally, the algorithm is capable of object tracking and Siteswap inference with up to 11 objects for a variety of challenging Siteswaps and conditions, serving as a versatile performance analysis, evaluation, and visualisation utility.

This paper investigates the task of tracking multiple objects in clutter under a distributed multi-sensor network with time-varying connectivity. Designed with the same objective as the centralised variational multi-object tracker, the proposed method achieves optimal decentralised fusion in performance with local processing and communication with only neighboring sensors. A key innovation is the decentralised construction of a locally maximised evidence lower bound, which greatly reduces the information required for communication. Our decentralised natural gradient descent variational multi-object tracker, enhanced with the gradient tracking strategy and natural gradients that adjusts the direction of traditional gradients to the steepest, shows rapid convergence. Our results verify that the proposed method is empirically equivalent to the centralised fusion in tracking accuracy, surpasses suboptimal fusion techniques with comparable costs, and achieves much lower communication overhead than the consensus-based variational multi-object tracker.

In this paper we introduce tracking models based on non-Gaussian continuous time stochastic processes with time-varying skewness. The idea behind this is that the skewness of the dynamical model may be able to model a propensity for an object to undergo manoeuvres of a particular type, for example velocities tending in a particular direction, but that these may change over time. This process is constructed based on a random series representation of conditionally Gaussian Lévy processes, which enables straightforward simulation of the models. We demonstrate the specific example of α-stable processes and find that such processes can capture abrupt changes owing to their heavy-tailed behaviour, and demonstrate the random changes in direction caused by the time-changing skewness of the distribution. We propose methods for joint tracking of both states and skewness for such processes, based on a marginalised particle filter, which are demonstrated to perform well even with limited numbers of particles.

A key challenge for multi-target trackers is being able to track both agile and simply moving targets effectively. This is only furthered by the standard difficulties of automated track initiation and deletion. This paper proposes a solution to this, where unknown associations and track management are handled by a Dirichlet process prior, and Gaussian processes model the dynamics of the targets. The promising performance of the proposed tracker is demonstrated on both synthetic and real radar data against a selection of other methods.

In this work we present the variance-gamma driven state-space model (VGSSM) - a linear vector stochastic differential equation driven by the variance-gamma (VG) Lévy process, and propose a novel inference framework in such systems. There are closed form expressions for the first four moments of the marginals of the VG process, allowing for more flexible modelling than Brownian motion (BM), retaining BM as a limiting case. The conditionally Gaussian formulation of the variance-gamma process lends itself well to the use of a marginalised particle filter (MPF) which can include the estimation of model parameters as part of the sampling framework. As an example we present a state-space formulation of Langevin dynamics in the VGSSM for estimation of both the observed and the latent first-order dynamics of a system. We apply this specific Langevin formulation to synthetically generated data to validate the results of the MPF, followed by an application to foreign-exchange tick data to demonstrate the method for trend tracking in data sets that are irregularly sampled in time.

This paper addresses the joint detection and tracking of an unknown and time-varying number of targets in clutter. Here we formulate the tracking task in a variable-dimension state space, under which the reversible jump sequential Markov chain Monte Carlo sampling methods can be utilised to online estimate the target number, their kinematic states, and the association variables. In particular, a fast Rao-Blackwellisation scheme is devised to improve the tracking accuracy and sampling efficiency for linear Gaussian models. Based on the nonhomogeneous Poisson process measurement model, the developed tracker enjoys a partially parallel sampling structure, thereby being able to efficiently tackle the data association under massive measurements and clutter. The simulation results demonstrate that the developed tracker exhibits superior tracking performance in comparison to existing trackers in both accuracy and computational efficiency when tracking multiple targets under heavy clutter.

In this paper we present on-line Bayesian filtering methods for non-linear multivariate time series models corrupted by generalised asymmetric Laplace noise. We derive the optimum kernel for a particle filter applied to multivariate non-linear state-space models with scalar observations, where the observation noise is additive and asymmetric Laplacian. We show that sampling from this multivariate kernel is tractable using commonly available methods for use in particle filters, and that its associated likelihood can be evaluated. A particle filter is implemented for a test case using the developed kernel, and its performance is compared to that of a traditional bootstrap filter. The proposed methods show potential for application to systems with heavy-tailed skew noise.

In this paper we present multidimensional α-stable state-space models for object tracking, expressed in continuous time as Lévy processes. In contrast with the conventional Gaussian models, these heavy-tailed α-stable models are more likely to exhibit extreme noise values, thus showing the capability for modeling of erratic manoeuvring behaviour. Despite the potential benefits, such models are usually highly intractable for inference and therefore have not yet been widely adopted in the tracking field. Here the models are represented in a conditionally Gaussian series form, so that the marginal (Rao-Blackwellised) particle filter can be employed to perform tracking and smoothing very efficiently. As the result, the simulation tracks present some sharp manoeuvres, owing to the heavy-tailed property, and experiments demonstrate improved performance on an intent inference problem from automotive UI with highly perturbed pointing data.

This paper introduces a novel leader-follower model for tracking a group of manoeuvring objects under a probabilistic framework. The proposed model develops on the conventional leader-follower model in which the followers are driven stochastically towards the velocity and position of the leader. Here we consider the dynamic of followers as a mean-reverting process and express it in a continuous-time stochastic differential equation. Instead of using a standard global Cartesian or polar system, an intrinsic coordinate model is utilised for the leader where piecewise constant forces are applied relative to the heading of the leader. Followers then mean revert towards the heading angle and speed of the leader, leading to a more realistic behavioural modelling than the more conventional global coordinate systems. Such a dynamical model is readily incorporated into tracking algorithms using for example the variable rate particle filtering framework which can accurately capture and estimate the manoeuvres of the leader and followers. The simulation results verify its efficacy under challenging group tracking scenarios and future work will explore automatic identification of group structure and leadership from measurements of groups of moving objects.

This paper introduces a Bayesian framework for estimating the probability of a driver or passenger(s) returning to the vehicle, from the available partial (noisy) track of his/her location. The latter can be provided by a smartphone navigational service and/or other dedicated user to vehicle positioning solution, for instance RF-based. The proposed approach treats the addressed intent prediction problem, i.e. not tracking the object's state (e.g. the driver/passenger position, velocity, etc.) or predicting its next few values, within an object tracking formulation, leading to a Kalman-filter-based implementation of the inference routine. Hence, it is dubbed meta-tracker in lieu of a conventional “sensor-level” tracking algorithm and relies on utilising bridging distributions to encapsulate the long term dependencies in the trajectory followed by the driver or passenger as dictated by the intended endpoint, if any. Two example trajectories are shown to demonstrate the effectiveness of this flexible framework.

In this paper we address the challenging task of tracking a fast-moving bicycle, in the indoor velodrome environment, using inertial sensors and infrequent position measurements. Since the inertial sensors are physically in the intrinsic frame of the bike, we adopt an intrinsic frame dynamic model for the motion, based on curvilinear dynamical models for manoeuvring objects. We show that the combination of inertial measurements with the intrinsic dynamic model leads to linear equations, which may be incorporated effectively into particle filtering schemes. Position measurements are provided through timing measurements on the track from a camera-based system and these are fused with the inertial measurements using a particle filter weighting scheme. The proposed methods are evaluated on synthesised cycling datasets based on real motion trajectories, showing their potential accuracy, and then real data experiments are reported.

This paper describes a study on modelling the Received Signal Strength Indicator (RSSI) measured by the smartphone of a vehicle user. The present transmissions are emitted by dedicated radio frequency sources, such as Bluetooth Low Energy (BLE) beacons, mounted to the vehicle to determine the driver/passenger(s) proximity or relative position(s). Based on empirical data, a model of the measurements noise, which utilises skewed distributions, is proposed to capture inconsistencies in reception and the impact of occlusions on the RSSI profile in an automotive setting, for example occlusions in car parks. Experimental data is used to demonstrate the suitability of the introduced model.

We propose a deterministic recursive algorithm for approximate Bayesian filtering. The proposed filter uses a function referred to as the approximate Gaussian flow transformation that transforms a Gaussian prior random variable into an approximate posterior random variable. Given a Gaussian filter prediction distribution, the succeeding filter prediction is approximated as Gaussian by applying sigma point moment-matching to the composition of the Gaussian flow transformation and the state transition function. This requires linearising the measurement model at each sigma point, solving the linearised models analytically, and introducing the measurement information gradually to improve the linearisation points progressively. Computer simulations show that the proposed method can provide higher accuracy and better posterior covariance matrix approximation than some state-of-the art computationally light approximative filters when the measurement model function is nonlinear but differentiable and the noises are additive and Gaussian. We also present a highly nonlinear scenario where the proposed filter occasionally diverges. In the accuracy-computational complexity axis the proposed algorithm is between Kalman filter extensions and Monte Carlo methods.

A model for vehicle motion on a road network Is developed using an enumeration of feasible routes. Combined with a generic stochastic model of distance travelled, a predicted pdf of vehicle position is derived as a mixture. This approach allows prior information on vehicle intent and behaviour to be included via the mixture weights. Illustrative examples are given using a second-order linear-Gaussian model for vehicle road speed. The value of road map data is shown via a tracking example with poor quality measurements and a substantial period prior to sensor activation. The tracking algorithm is implemented using a standard particle filter. In particular, the scheme has potential for revealing the likely paths taken by the vehicle.