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

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.

💡 Research Summary

The paper introduces the Trajectory‑Information Exchange Multi‑Bernoulli (T‑IEMB) filter, a novel Bayesian solution for track‑before‑detect (TbD) scenarios that simultaneously estimates the set of currently alive trajectories and the set of all trajectories (including those that have already terminated). The authors first define a generalized superpositional measurement model. In this model each target’s state is first transformed by internal nonlinear functions h(·) (producing a hidden measurement) and R(·) (producing a hidden covariance). The contributions of all targets are summed, and then external mapping functions m(·) and Σ(·) convert the summed hidden quantities into the conditional mean μ_k and covariance Σ_k of the raw sensor measurement. This formulation captures a broad class of non‑Gaussian measurement distributions (e.g., Rayleigh, Rice, K‑distribution) while retaining only the first two conditional moments for filter design.

To exploit this model, the authors propose two Gaussian implementations for approximating μ_k and Σ_k. The first uses a first‑order Taylor linearisation of the external mappings; the second employs the Iterated Posterior Linearisation Filter (IPLF), which iteratively refines a linear approximation of the nonlinear mappings based on the posterior. Both approaches avoid the high‑dimensional sampling required by particle filters, yet they provide accurate moment estimates needed for a Gaussian update.

The T‑IEMB filter extends the Information Exchange Multi‑Bernoulli (IEMB) filter from point‑target sets to trajectory sets. Two trajectory‑Multi‑Bernoulli (TMB) densities are defined: TMB_alive, which contains only trajectories that are still present at the current time step, and TMB_all, which also retains trajectories that have already died. For alive trajectories, survival probability p_S(x_v) and a single‑trajectory transition density g(·) drive the prediction step; dead trajectories are kept unchanged in TMB_all. The prediction‑update recursion follows the standard RFS Bayes filter, but the update step is performed by minimizing the Kullback‑Leibler divergence (KLD) between the exact posterior (which is not a Multi‑Bernoulli) and a tractable Multi‑Bernoulli approximation. This KLD minimisation is carried out after introducing auxiliary variables, exactly as in the original IEMB derivation, ensuring that information exchange among individual Bernoulli components correctly accounts for the superpositional coupling in the measurement.

Simulation results focus on a radar‑based TbD scenario with non‑Gaussian (Rician) measurement noise. Two Gaussian T‑IEMB variants (Taylor‑linearised and IPLF‑based) are benchmarked against a state‑of‑the‑art particle‑filter implementation of the superpositional PHD/MB filter. Performance is evaluated using the OSPA‑2 metric for both alive‑trajectory and all‑trajectory sets. The Gaussian T‑IEMB filters achieve 12–18 % lower OSPA‑2 errors than the particle filter, indicating more accurate trajectory reconstruction. Moreover, computational time is reduced by a factor of three or more, demonstrating that the proposed approach is suitable for real‑time applications.

In summary, the paper makes four key contributions: (1) a generalized superpositional measurement model that unifies many non‑Gaussian sensor models; (2) a practical Gaussian moment‑approximation scheme (Taylor and IPLF) that eliminates the need for costly particle representations; (3) an extension of the IEMB framework to trajectory‑level inference, yielding the T‑IEMB filter capable of estimating both alive and all trajectories; and (4) empirical evidence that the proposed filter outperforms existing particle‑based methods in accuracy while being computationally lighter. These advances open the door to efficient, high‑fidelity multi‑target tracking in low‑SNR environments such as radar surveillance, underwater acoustics, and other TbD applications.