  The probability hypothesis density (PHD) and Poisson multi-Bernoulli (PMB)
filters are two popular set-type multi-object filters. Motivated by the fact
that the multi-object filtering density after each update step in the PHD
filter is a PMB without approximation, in this paper we present a multi-object
smoother involving PHD forward filtering and PMB backward smoothing. This is
achieved by first running the PHD filtering recursion in the forward pass and
extracting the PMB filtering densities after each update step before the
Poisson Point Process approximation, which is inherent in the PHD filter
update. Then in the backward pass we apply backward simulation for sets of
trajectories to the extracted PMB filtering densities. We call the resulting
multi-object smoother hybrid PHD-PMB trajectory smoother. Notably, the hybrid
PHD-PMB trajectory smoother can provide smoothed trajectory estimates for the
PHD filter without labeling or tagging, which is not possible for existing PHD
smoothers. Also, compared to the trajectory PHD filter, which can only estimate
alive trajectories, the hybrid PHD-PMB trajectory smoother enables the
estimation of the set of all trajectories. Simulation results demonstrate that
the hybrid PHD-PMB trajectory smoother outperforms the PHD filter in terms of
both state and cardinality estimates, and the trajectory PHD filter in terms of
false detections.
