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3D Trajectory Reconstruction of Moving Points Based on a Monocular Camera

Huayu Huang, Banglei Guan, Yang Shang, Qifeng Yu

TL;DR

This work addresses monocular 3D trajectory reconstruction by representing moving point motion with temporal polynomials and mitigating ill-conditioning via ridge estimation. It introduces an automatic order selection mechanism and a reconstructability metric to quantify and improve accuracy under limited observations. The approach outperforms DCT-based and TI-based baselines in simulations and real UAV data, especially when observations are scarce or noisy. The method offers a practical, efficient solution for high-precision target motion reconstruction with a single camera, with implications for UAV sensing and navigation.

Abstract

The motion measurement of point targets constitutes a fundamental problem in photogrammetry, with extensive applications across various engineering domains. Reconstructing a point's 3D motion just from the images captured by only a monocular camera is unfeasible without prior assumptions. Under limited observation conditions such as insufficient observations, long distance, and high observation error of platform, the least squares estimation faces the issue of ill-conditioning. This paper presents an algorithm for reconstructing 3D trajectories of moving points using a monocular camera. The motion of the points is represented through temporal polynomials. Ridge estimation is introduced to mitigate the issues of ill-conditioning caused by limited observation conditions. Then, an automatic algorithm for determining the order of the temporal polynomials is proposed. Furthermore, the definition of reconstructability for temporal polynomials is proposed to describe the reconstruction accuracy quantitatively. The simulated and real-world experimental results demonstrate the feasibility, accuracy, and efficiency of the proposed method.

3D Trajectory Reconstruction of Moving Points Based on a Monocular Camera

TL;DR

This work addresses monocular 3D trajectory reconstruction by representing moving point motion with temporal polynomials and mitigating ill-conditioning via ridge estimation. It introduces an automatic order selection mechanism and a reconstructability metric to quantify and improve accuracy under limited observations. The approach outperforms DCT-based and TI-based baselines in simulations and real UAV data, especially when observations are scarce or noisy. The method offers a practical, efficient solution for high-precision target motion reconstruction with a single camera, with implications for UAV sensing and navigation.

Abstract

The motion measurement of point targets constitutes a fundamental problem in photogrammetry, with extensive applications across various engineering domains. Reconstructing a point's 3D motion just from the images captured by only a monocular camera is unfeasible without prior assumptions. Under limited observation conditions such as insufficient observations, long distance, and high observation error of platform, the least squares estimation faces the issue of ill-conditioning. This paper presents an algorithm for reconstructing 3D trajectories of moving points using a monocular camera. The motion of the points is represented through temporal polynomials. Ridge estimation is introduced to mitigate the issues of ill-conditioning caused by limited observation conditions. Then, an automatic algorithm for determining the order of the temporal polynomials is proposed. Furthermore, the definition of reconstructability for temporal polynomials is proposed to describe the reconstruction accuracy quantitatively. The simulated and real-world experimental results demonstrate the feasibility, accuracy, and efficiency of the proposed method.

Paper Structure

This paper contains 13 sections, 23 equations, 17 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Linear reconstruction of a 3D point trajectory
  4. Trajectory reconstruction under ridge estimation
  5. Automatic selection method for the order parameter
  6. 3D trajectory reconstructability analysis
  7. Simulated experiments
  8. Feasibility
  9. Accuracy and robustness
  10. Performance of the $K$ selection method
  11. Performance of handling missing data
  12. Real-world experiments
  13. Conclusion

Figures (17)

  • Figure 1: 3D reconstruction of a point based on a monocular camera. (a) A static point. (b) A moving point.
  • Figure 2: Illustration of 3D trajectory reconstruction of a moving target based on a flight platform equipped with a monocular camera.
  • Figure 3: Illustration of residual error.
  • Figure 4: Imaging maps of different inclination angles.
  • Figure 5: Degeneracy situations. (a) The order of the temporal polynomial representing the camera motion is lower than that of the target's motion. (b) All sight-rays intersect at the same point. (c) All sight-rays are parallel.
  • ...and 12 more figures