  In this paper we introduce spatiotemporal constraints for trajectories, i.e.,
restrictions that the trajectory must be in some part of the state space
(spatial constraint) at some point in time (temporal constraint).
Spatiotemporal contraints on trajectories can be used to answer a range of
important questions, including, e.g., "where did the person that were in area A
at time t, go afterwards?". We discuss how multiple constraints can be combined
into sets of constraints, and we then apply sets of constraints to set of
trajectories densities, specifically Poisson Multi-Bernoulli Mixture (PMBM)
densities. For Poisson target birth, the exact posterior density is PMBM for
both point targets and extended targets. In the paper we show that if the
unconstrained set of trajectories density is PMBM, then the constrained density
is also PMBM. Examples of constrained trajectory densities motivate and
illustrate the key results.
