We provide a derivation of the Poisson multi-Bernoulli mixture (PMBM) filter for multi-target tracking with the standard point target measurements without using probability generating functionals or functional derivatives. We also establish the connection with the δ-generalised labelled multi-Bernoulli (δ-GLMB) filter, showing that a δ-GLMB density represents a multi-Bernoulli mixture with labelled targets so it can be seen as a special case of PMBM. In addition, we propose an implementation for linear/Gaussian dynamic and measurement models and how to efficiently obtain typical estimators in the literature from the PMBM. The PMBM filter is shown to outperform other filters in the literature in a challenging scenario.

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This paper presents a Poisson multi-Bernoulli mixture (PMBM) conjugate prior for multiple extended object filtering. A Poisson point process is used to describe the existence of yet undetected targets, while a multi-Bernoulli mixture describes the distribution of the targets that have been detected. The prediction and update equations are presented for the standard transition density and measurement likelihood. Both the prediction and the update preserve the PMBM form of the density, and in this sense the PMBM density is a conjugate prior. However, the unknown data associations lead to an intractably large number of terms in the PMBM density, and approximations are necessary for tractability. A gamma Gaussian inverse Wishart implementation is presented, along with methods to handle the data association problem. A simulation study shows that the extended target PMBM filter performs well in comparison to the extended target δ-GLMB and LMB filters. An experiment with Lidar data illustrates the benefit of tracking both detected and undetected targets.

This paper is concerned with Gaussian approximations to the posterior probability density function (PDF) in the update step of Bayesian filtering with nonlinear measurements. In this setting, sigma-point approximations to the Kalman filter (KF) recursion are widely used due to their ease of implementation and relatively good performance. In the update step, these sigma-point KFs are equivalent to linearising the nonlinear measurement function by statistical linear regression (SLR) with respect to the prior PDF.In this paper, we argue that the measurement function should be linearised using SLR with respect to the posterior rather than the prior to take into account the information provided by the measurement. The resulting filter is referred to as the posterior linearisation filter (PLF). In practice, the exact PLF update is intractable but can be approximated by the iterated PLF (IPLF), which carries out iterated SLRs with respect to the best available approximation to the posterior. The IPLF can be seen as an approximate recursive Kullback-Leibler divergence minimisation procedure. We demonstrate the high performance of the IPLF in relation to other Gaussian filters in two numerical examples.