Jacobian Computation for Cumulative B-Splines on SE(3) and Application to Continuous-Time Object Tracking

TL;DR

This paper introduces a novel continuous-time 6-DoF object tracking method using cumulative B-Splines on SE(3), with analytical Jacobians that improve computational efficiency and velocity estimation accuracy.

Contribution

It presents the first continuous-time 6-DoF object tracking approach with analytical Jacobians for SE(3), enabling efficient interpolation of poses and velocities from RGB-D data.

Findings

Achieves competitive localization results on benchmarks.

Significantly improves velocity estimation accuracy.

Reduces computational cost through analytical Jacobians.

Abstract

In this paper we propose a method that estimates the $SE(3)$ continuous trajectories (orientation and translation) of the dynamic rigid objects present in a scene, from multiple RGB-D views. Specifically, we fit the object trajectories to cumulative B-Splines curves, which allow us to interpolate, at any intermediate time stamp, not only their poses but also their linear and angular velocities and accelerations. Additionally, we derive in this work the analytical $SE(3)$ Jacobians needed by the optimization, being applicable to any other approach that uses this type of curves. To the best of our knowledge this is the first work that proposes 6-DoF continuous-time object tracking, which we endorse with significant computational cost reduction thanks to our analytical derivations. We evaluate our proposal in synthetic data and in a public benchmark, showing competitive results in localization and significant improvements in velocity estimation in comparison to discrete-time approaches.