  This paper presents the probability hypothesis density filter (PHD) and the
cardinality PHD (CPHD) filter for sets of trajectories, which are referred to
as the trajectory PHD (TPHD) and trajectory CPHD (TCPHD) filters. Contrary to
the PHD/CPHD filters, the TPHD/TCPHD filters are able to produce trajectory
estimates from first principles. The TPHD filter is derived by recursively
obtaining the best Poisson multitrajectory density approximation to the
posterior density over the alive trajectories by minimising the
Kullback-Leibler divergence. The TCPHD is derived in the same way but
propagating an independent identically distributed (IID) cluster
multitrajectory density approximation. We also propose the Gaussian mixture
implementations of the TPHD and TCPHD recursions, the Gaussian mixture TPHD
(GMTPHD) and the Gaussian mixture TCPHD (GMTCPHD), and the L-scan
computationally efficient implementations, which only update the density of the
trajectory states of the last L time steps.
