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

arXiv: https://arxiv.org/abs/2201.10602 · HTML: https://mlsanity.com/paper/2201.10602/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 54 references that are in the map.
- First posted on arXiv 25 Jan 2022.

## Builds on (most cited first, 19 of 19)

- [Mask R-CNN](https://mlsanity.com/paper/1703.06870/) (2017)
- [Past, Present, and Future of Simultaneous Localization And Mapping: Towards the Robust-Perception Age](https://mlsanity.com/paper/1606.05830/) (2016)
- [Fast Online Object Tracking and Segmentation: A Unifying Approach](https://mlsanity.com/paper/1812.05050/) (2018)
- [DynaSLAM: Tracking, Mapping and Inpainting in Dynamic Scenes](https://mlsanity.com/paper/1806.05620/) (2018)
- [A micro Lie theory for state estimation in robotics](https://mlsanity.com/paper/1812.01537/) (2018)
- [CubeSLAM: Monocular 3D Object SLAM](https://mlsanity.com/paper/1806.00557/) (2018)
- [Continuous-Time Visual-Inertial Odometry for Event Cameras](https://mlsanity.com/paper/1702.07389/) (2017)
- [DynaSLAM II: Tightly-Coupled Multi-Object Tracking and SLAM](https://mlsanity.com/paper/2010.07820/) (2020)
- [VDO-SLAM: A Visual Dynamic Object-aware SLAM System](https://mlsanity.com/paper/2005.11052/) (2020)
- [Efficient Derivative Computation for Cumulative B-Splines on Lie Groups](https://mlsanity.com/paper/1911.08860/) (2019)
- [Stereo Vision-based Semantic 3D Object and Ego-motion Tracking for Autonomous Driving](https://mlsanity.com/paper/1807.02062/) (2018)
- [Efficient Continuous-Time SLAM for 3D Lidar-Based Online Mapping](https://mlsanity.com/paper/1810.06802/) (2018)
- [ClusterVO: Clustering Moving Instances and Estimating Visual Odometry for Self and Surroundings](https://mlsanity.com/paper/2003.12980/) (2020)
- [DOT: Dynamic Object Tracking for Visual SLAM](https://mlsanity.com/paper/2010.00052/) (2020)
- [The Oxford Multimotion Dataset: Multiple SE(3) Motions with Ground Truth](https://mlsanity.com/paper/1901.01445/) (2019)
- [Asynchronous Multi-View SLAM](https://mlsanity.com/paper/2101.06562/) (2021)
- [Multimotion Visual Odometry (MVO): Simultaneous Estimation of Camera and Third-Party Motions](https://mlsanity.com/paper/1808.00274/) (2018)
- [Trajectory Representation and Landmark Projection for Continuous-Time Structure from Motion](https://mlsanity.com/paper/1805.02543/) (2018)
- [Multimotion Visual Odometry (MVO)](https://mlsanity.com/paper/2110.15169/) (2021)

## Abstract (arXiv)

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.