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The Mean of Multi-Object Trajectories

Tran Thien Dat Nguyen, Ba Tuong Vo, Ba-Ngu Vo, Hoa Van Nguyen, Changbeom Shim

TL;DR

The Fréchet mean, and metrics based on the optimal sub-pattern assignment (OSPA) construct, are used to extend the notion of average from vectors to trajectories and multi-object trajectories.

Abstract

This paper introduces the concept of a mean for trajectories and multi-object trajectories (defined as sets or multi-sets of trajectories) along with algorithms for computing them. Specifically, we use the Fréchet mean, and metrics based on the optimal sub-pattern assignment (OSPA) construct, to extend the notion of average from vectors to trajectories and multi-object trajectories. Further, we develop efficient algorithms to compute these means using greedy search and Gibbs sampling. Using distributed multi-object tracking as an application, we demonstrate that the Fréchet mean approach to multi-object trajectory consensus significantly outperforms state-of-the-art distributed multi-object tracking methods.

The Mean of Multi-Object Trajectories

TL;DR

The Fréchet mean, and metrics based on the optimal sub-pattern assignment (OSPA) construct, are used to extend the notion of average from vectors to trajectories and multi-object trajectories.

Abstract

This paper introduces the concept of a mean for trajectories and multi-object trajectories (defined as sets or multi-sets of trajectories) along with algorithms for computing them. Specifically, we use the Fréchet mean, and metrics based on the optimal sub-pattern assignment (OSPA) construct, to extend the notion of average from vectors to trajectories and multi-object trajectories. Further, we develop efficient algorithms to compute these means using greedy search and Gibbs sampling. Using distributed multi-object tracking as an application, we demonstrate that the Fréchet mean approach to multi-object trajectory consensus significantly outperforms state-of-the-art distributed multi-object tracking methods.
Paper Structure (33 sections, 7 theorems, 67 equations, 7 figures, 4 tables, 7 algorithms)

This paper contains 33 sections, 7 theorems, 67 equations, 7 figures, 4 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Distributed Multi-Object Estimation
  4. Metrics for Point Patterns and Trajectories
  5. Trajectory and Fréchet Mean
  6. The Fréchet Mean Trajectory
  7. Computing the OSPA Fréchet Mean Trajectory
  8. Problem Transformation
  9. Mean Trajectory via Greedy Search
  10. Multi-Object Trajectory and Fréchet Mean
  11. The Fréchet Mean Multi-Object Trajectory
  12. Computing the Fréchet Mean Multi-Object Trajectory
  13. Problem Transformation
  14. Mean Multi-Object Trajectory via Greedy Search
  15. Fréchet Mean via Gibbs Sampling
  16. ...and 18 more sections

Key Result

Proposition 1

Suppose $\hat{v}$ is an $r^{th}$-order Fréchet mean of the trajectories $v^{(1:N)}$. Then for each $k\in\mathcal{D}_{\hat{v}}$, $\hat{v}(k)$ is the $r^{th}$-order Fréchet mean of the trajectory states $v^{(n)}(k)$ that exist at time $k$, weighted by $|\mathcal{D}_{\hat{v}}\cup\mathcal{D}_{v^{(n)}}|$

Figures (7)

  • Figure 1: Two examples of the mean trajectory (thick black) of multiple input trajectories (color). Marker sizes indicate times, with the same size representing the same time, and sizes decrease toward the ends of trajectories. Consecutive instances of a trajectory are connected with solid lines.
  • Figure 2: Examples of the mean multi-object trajectory (MOT) of three sample multi-object trajectories. Each trajectory in a multi-object trajectory has a distinct color. Consecutive instances of a trajectory are connected with solid lines. Intuitively, the mean tends to smooth out the data points.
  • Figure 3: True trajectories and network configuration where the nodes are annotated by [ifgeo]96, or ♦, see Table \ref{['tab:sensor-nodes']} for details of the sensor nodes.
  • Figure 4: Sample outputs from FM(1) with Euclidean distance and greedy search, TC and DBSCAN-T.
  • Figure 5: Sample output from FM(1) fusion with uncertainty. The ellipses represent 95% confidence intervals of the Gaussian distributions. Dashed line ellipses are estimates from the nodes and solid ellipses are the fused results.
  • ...and 2 more figures

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Proposition 1
  • Lemma 1
  • Proposition 2
  • Proposition 3
  • Definition 4
  • Remark 2
  • ...and 7 more