Continuous-Discrete Multiple Target Filtering: PMBM, PHD and CPHD Filter Implementations

This article develops models and algorithms for continuous-discrete multiple target filtering, in which the multi-target system is modelled in continuous time and measurements are available at discrete time steps. In order to do so, this paper first proposes a statistical model for multi-target appearance, dynamics and disappearance in continuous time, based on continuous time birth/death processes and stochastic differential equations. The multitarget state is observed at known time instants based on the standard measurement model, and the objective is to compute the distribution of the multi-target state at these time steps. For the Wiener velocity model, we derive a closed-form formula to obtain the best Gaussian Poisson point process fit to the birth density based on Kullback-Leibler minimisation. The resulting discretised model gives rise to the continuous-discrete Gaussian Poisson multi-Bernoulli mixture (PMBM) filter, the continuous-discrete Gaussian mixture probability hypothesis density (PHD) filter and the continuous-discrete Gaussian mixture cardinality PHD (CPHD) filter. The proposed filters are specially useful for multi-target estimation when the time interval between measurements is non-uniform.