# Trajectory PHD and CPHD filters

arXiv: https://arxiv.org/abs/1811.08820 · HTML: https://mlsanity.com/paper/1811.08820/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Nov 2018.

## Built on by (most cited first, 5 of 5)

- [GNN-PMB: A Simple but Effective Online 3D Multi-Object Tracker without Bells and Whistles](https://mlsanity.com/paper/2206.10255/) (2022)
- [Transformer-Based Multi-Object Smoothing with Decoupled Data Association and Smoothing](https://mlsanity.com/paper/2312.17261/) (2023)
- [Continuous-discrete multiple target tracking with out-of-sequence measurements](https://mlsanity.com/paper/2106.04898/) (2021)
- [Extended Object Tracking Using Sets Of Trajectories with a PHD Filter](https://mlsanity.com/paper/2109.01019/) (2021)
- [Poisson multi-Bernoulli mixture filter for trajectory measurements](https://mlsanity.com/paper/2504.08421/) (2025)

## Abstract (arXiv)

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.