A Track-Before-Detect Trajectory Multi-Bernoulli Filter for Generalised Superpositional Measurements

TL;DR

The paper tackles track-before-detect TkBD in multi-target scenes using generalised superpositional measurements by extending the Information Exchange Multi-Bernoulli (IEMB) framework to sets of trajectories, yielding the Trajectory-IEMB (T-IEMB) filter. A Gaussian implementation (GT-IEMB) is developed, with an Iterated Posterior Linearisation-based (IPLF) update to handle nonlinear measurement moments, enabling efficient fusion of trajectory information. The authors introduce both alive and all trajectory variants, derive prediction and update recursions for trajectory MB densities, and provide practical Gaussian updates that avoid particle methods. Simulation results in nonGaussian TkBD scenarios show GT-IEMB with IPLF outperforming a state-of-the-art particle-filter baseline (GPP-MB) while incurring much lower computational cost, and demonstrate gains from increasing the L-scan window up to a point. The work advances robust trajectory-level TkBD tracking and broadens applicability to non-Gaussian measurement models such as Rician/Rayleigh clutter, with potential extensions to real data and SMC-based implementations.

Abstract

This paper proposes the Trajectory-Information Exchange Multi-Bernoulli (T-IEMB) filter to estimate sets of alive and all trajectories in track-before-detect applications with generalised superpositional measurements. This measurement model has superpositional hidden variables which are mapped to the conditional mean and covariance of the measurement, enabling it to describe a broad range of measurement models. This paper also presents a Gaussian implementation of the T-IEMB filter, which performs the update by approximating the conditional moments of the measurement model, and admits a computationally light filtering solution. Simulation results for a non-Gaussian radar-based tracking scenario demonstrate the performance of two Gaussian T-IEMB implementations, which provide improved tracking performance compared to a state-of-the-art particle filter based solution for track-before-detect, at a reduced computational cost.

Figures (4)

• Figure 1: Graphical representation of how $n$ targets $\mathbf{x}_{k}=\{x_{k}^{1},...,x_{k}^{n}\}$ at time step $k$ contribute to the measurement through each hidden measurement function $h(\cdot)$ and covariance function $R(\cdot)$, which are summed over all targets and passed through the mapping functions $m(\cdot)$ and $\Sigma(\cdot)$, providing the conditional mean and covariance of the measurement.
• Figure 2: T-IEMB filter diagram. A Bayes update of the predicted TMB density and considered likelihood yields a general non-TMB density. An approximate updated TMB density is derived by minimising the KLD (after the introduction of the auxiliary variables) Davies2024.
• Figure 3: Ground truth trajectories of each target, where targets 1 through 4 are born at time steps 3, 16, 17, and 20, respectively, and die at time steps 74, 64, 57 and 64, respectively. Each radar resolution cell is of dimensions $10\ \textrm{m}\times10\ \textrm{m}$, which is illustrated using the grid.
• Figure 4: Total RMS T-GOSPA cost of each filter for the set of alive trajectories. L$x$ denotes a filter with an $L$-scan length of $x$.

Theorems & Definitions (7)

• Example 1
• Lemma 2: T-IEMB Prediction
• Lemma 3: T-IEMB Update for Alive Trajectories
• Lemma 4: T-IEMB Update for All Trajectories
• Lemma 5
• Lemma 6
• Proposition 7