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Computer Vision Group
TUM Department of Informatics
Technical University of Munich

Technical University of Munich



Efficient Derivative Computation for Cumulative B-Splines on Lie Groups


Continuous-time trajectory representation has recently gained popularity for tasks where the fusion of high-frame-rate sensors and multiple unsynchronized devices is required. Lie group cumulative B-splines are a popular way of representing continuous trajectories without singularities. They have been used in near real-time SLAM and odometry systems with IMU, LiDAR, regular, RGB-D and event cameras, as well as for offline calibration.

These applications require efficient computation of time derivatives (velocity, acceleration), but all prior works rely on a computationally suboptimal formulation. In this work we present an alternative derivation of time derivatives based on recurrence relations that needs O(k) instead of O(k^2) matrix operations (for a spline of order k) and results in simple and elegant expressions. While producing the same result, the proposed approach significantly speeds up the trajectory optimization and allows for computing simple analytic derivatives with respect to spline knots. The results presented in this paper pave the way for incorporating continuous-time trajectory representations into more applications where real-time performance is required.



Open-Source Code

The code for the experiments presented in the paper is available at https://gitlab.com/tum-vision/lie-spline-experiments.

If you are planning to use the code in your project check the 'include/basalt/spline' folder of the headers-only library. ( Documentation)

For a calibration tool based on the proposed B-spline trajectory representation check the dataset and device calibration tutorials of the basalt project.

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Conference and Workshop Papers
[]Efficient Derivative Computation for Cumulative B-Splines on Lie Groups (C. Sommer, V. Usenko, D. Schubert, N. Demmel and D. Cremers), In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2020.  [bibtex] [doi] [arXiv:1911.08860] [pdf]Oral Presentation
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