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From Target Tracking to Targeting Track -- Part I: A Metric for Spatio-Temporal Trajectory Evaluation

Tiancheng Li, Yan Song, Hongqi Fan, Jingdong Chen

TL;DR

This work introduces Star-ID, a spatio-temporal distance for comparing continuous-time Trajectory Functions of Time (T-FoTs) in target tracking. By aligning FoTs in the spatio-temporal domain and distinguishing temporally aligned versus unaligned segments, Star-ID integrates both localization errors and false/missed trajectory penalties, with a time-averaged variant for online use. The main result establishes that Star-ID is a proper distance under natural penalty settings, and the paper provides interpretation of its parameters and a detailed simulation study. Together with companion parts that model trajectory uncertainty via stochastic processes and online learning, Star-ID offers a principled framework for evaluating and improving continuous-time tracking performance in single and multi-target scenarios.

Abstract

In the realm of target tracking, performance evaluation plays a pivotal role in the design, comparison, and analytics of trackers. Compared with the traditional trajectory composed of a set of point-estimates obtained by a tracker in the measurement time-series, the trajectory that our series of studies including this paper pursued is given by a curve function of time (FoT). The trajectory FoT provides complete information of the movement of the target over time and can be used to infer the state corresponding to arbitrary time, not only at the measurement time. However, there are no metrics available for comparing and evaluating the trajectory FoT. To address this lacuna, we propose a metric denominated as the spatiotemporal-aligned trajectory integral distance (Star-ID). The StarID associates and aligns the estimated and actual trajectories in the spatio-temporal domain and distinguishes between the time-aligned and unaligned segments in calculating the spatial divergence including false alarm, miss-detection and localization errors. The effectiveness of the proposed distance metric and the time-averaged version is validated through theoretical analysis and numerical examples of a single target or multiple targets.

From Target Tracking to Targeting Track -- Part I: A Metric for Spatio-Temporal Trajectory Evaluation

TL;DR

This work introduces Star-ID, a spatio-temporal distance for comparing continuous-time Trajectory Functions of Time (T-FoTs) in target tracking. By aligning FoTs in the spatio-temporal domain and distinguishing temporally aligned versus unaligned segments, Star-ID integrates both localization errors and false/missed trajectory penalties, with a time-averaged variant for online use. The main result establishes that Star-ID is a proper distance under natural penalty settings, and the paper provides interpretation of its parameters and a detailed simulation study. Together with companion parts that model trajectory uncertainty via stochastic processes and online learning, Star-ID offers a principled framework for evaluating and improving continuous-time tracking performance in single and multi-target scenarios.

Abstract

In the realm of target tracking, performance evaluation plays a pivotal role in the design, comparison, and analytics of trackers. Compared with the traditional trajectory composed of a set of point-estimates obtained by a tracker in the measurement time-series, the trajectory that our series of studies including this paper pursued is given by a curve function of time (FoT). The trajectory FoT provides complete information of the movement of the target over time and can be used to infer the state corresponding to arbitrary time, not only at the measurement time. However, there are no metrics available for comparing and evaluating the trajectory FoT. To address this lacuna, we propose a metric denominated as the spatiotemporal-aligned trajectory integral distance (Star-ID). The StarID associates and aligns the estimated and actual trajectories in the spatio-temporal domain and distinguishes between the time-aligned and unaligned segments in calculating the spatial divergence including false alarm, miss-detection and localization errors. The effectiveness of the proposed distance metric and the time-averaged version is validated through theoretical analysis and numerical examples of a single target or multiple targets.

Paper Structure

This paper contains 25 sections, 3 theorems, 34 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Relevant Work
  3. Introduction to Companion Papers
  4. Contribution and Organization of This Paper
  5. Preliminaries
  6. OSPA: Metric on Sets of Points
  7. Unnormalized OSPA and GOSPA
  8. $\text{OSPA}^{(2)}$: Metric on Sets of Tracks
  9. Integral Multi-target Trajectory Assignment (IMTA) Metric
  10. Elemental Segment Distance
  11. Temporally Aligned Segment
  12. Temporally Unaligned Segment: SFA/SMD
  13. Distance between Two T-FoTs
  14. Proposed Metric: Star-ID
  15. Main Result
  16. ...and 10 more sections

Key Result

Theorem 1

When $c_{\text{SFA}}=c_{\text{SMD}} >0$ and $1\le p< \infty$, $d_{\boldsymbol{c}}^{(p)}\left( f,g \right)$ is a distance.

Figures (10)

  • Figure 1: Different tracking scenarios: (a) single target point-state estimation; (b) multiple target point-state estimation; (c) multi-target T-FoT estimation.
  • Figure 2: Two different T-FoTs share the same discretizing points.
  • Figure 3: Two T-FoT estimates overlap spatially but not temporally.
  • Figure 4: A target exists less than a sampling period and is undetected by the estimator.
  • Figure 5: Flowchart for calculating the Star-ID metric.
  • ...and 5 more figures

Theorems & Definitions (13)

  • Definition 1: $\ell_p$ distance
  • Definition 2: Distance metric
  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2
  • Definition 3: Time-averaged Star-ID
  • Definition 3: Time-averaged Star-ID
  • ...and 3 more