# Trajectory Representation and Landmark Projection for Continuous-Time Structure from Motion

arXiv: https://arxiv.org/abs/1805.02543 · HTML: https://mlsanity.com/paper/1805.02543/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 May 2018.

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

- [Asynchronous Multi-View SLAM](https://mlsanity.com/paper/2101.06562/) (2021)
- [Continuous-Time vs. Discrete-Time Vision-based SLAM: A Comparative Study](https://mlsanity.com/paper/2202.08894/) (2022)
- [EF-Calib: Spatiotemporal Calibration of Event- and Frame-Based Cameras Using Continuous-Time Trajectories](https://mlsanity.com/paper/2405.17278/) (2024)
- [Continuous-Time Spatiotemporal Calibration of a Rolling Shutter Camera-IMU System](https://mlsanity.com/paper/2108.07200/) (2021)
- [Jacobian Computation for Cumulative B-Splines on SE(3) and Application to Continuous-Time Object Tracking](https://mlsanity.com/paper/2201.10602/) (2022)

## Abstract (arXiv)

This paper revisits the problem of continuous-time structure from motion, and introduces a number of extensions that improve convergence and efficiency. The formulation with a C2-continuous spline for the trajectory naturally incorporates inertial measurements, as derivatives of the sought trajectory. We analyse the behaviour of split interpolation on mathbbSO(3) and on ℝ3, and a joint interpolation on mathbbSE(3), and show that the latter implicitly couples the direction of translation and rotation. Such an assumption can make good sense for a camera mounted on a robot arm, but not for hand-held or body-mounted cameras. Our experiments show that split interpolation on mathbbSO(3) and on ℝ3 is preferable over mathbbSE(3) interpolation in all tested cases. Finally, we investigate the problem of landmark reprojection on rolling shutter cameras, and show that the tested reprojection methods give similar quality, while their computational load varies by a factor of 2.