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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 ofimportant questions, including, e.g., "where did the person that were in area Aat time t, go afterwards?". We discuss how multiple constraints can be combinedinto sets of constraints, and we then apply sets of constraints to set oftrajectories densities, specifically Poisson Multi-Bernoulli Mixture (PMBM)densities. For Poisson target birth, the exact posterior density is PMBM forboth point targets and extended targets. In the paper we show that if theunconstrained set of trajectories density is PMBM, then the constrained densityis also PMBM. Examples of constrained trajectory densities motivate andillustrate the key results.

Spatiotemporal Constraints for Sets of Trajectories with Applications to PMBM Densities