# Spatiotemporal Constraints for Sets of Trajectories with Applications to PMBM Densities

arXiv: https://arxiv.org/abs/2002.12696 · HTML: https://mlsanity.com/paper/2002.12696/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Feb 2020.

## Built on by (most cited first, 1 of 1)

- [Continuous-discrete multiple target tracking with out-of-sequence measurements](https://mlsanity.com/paper/2106.04898/) (2021)

## Abstract (arXiv)

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.