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Computer Vision Group
TUM Department of Informatics
Technical University of Munich

Technical University of Munich



Square Root Marginalization for Sliding-Window Bundle Adjustment


In this paper we propose a novel square root sliding-window bundle adjustment suitable for real-time odometry applications. The square root formulation pervades three major aspects of our optimization-based sliding-window estimator: for bundle adjustment we eliminate landmark variables with nullspace projection; to store the marginalization prior we employ a matrix square root of the Hessian; and when marginalizing old poses we avoid forming normal equations and update the square root prior directly with a specialized QR decomposition. We show that the proposed square root marginalization is algebraically equivalent to the conventional use of Schur complement (SC) on the Hessian. Moreover, it elegantly deals with rank-deficient Jacobians producing a prior equivalent to SC with Moore–Penrose inverse. Our evaluation of visual and visual-inertial odometry on real-world datasets demonstrates that the proposed estimator is 36% faster than the baseline. It furthermore shows that in single precision, conventional Hessian-based marginalization leads to numeric failures and reduced accuracy. We analyse numeric properties of the marginalization prior to explain why our square root form does not suffer from the same effect and therefore entails superior performance.


Open-Source Code

The code is available on the master branch of Basalt. There is a tutorial on how to reproduce the experiments from the paper.

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Conference and Workshop Papers
[]Square Root Marginalization for Sliding-Window Bundle Adjustment (N Demmel, D Schubert, C Sommer, D Cremers and V Usenko), In IEEE International Conference on Computer Vision (ICCV), 2021. ([project page]) [bibtex] [arXiv:2109.02182] [pdf]
[]Square Root Bundle Adjustment for Large-Scale Reconstruction (N Demmel, C Sommer, D Cremers and V Usenko), In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2021. ([project page]) [bibtex] [arXiv:2103.01843] [pdf]
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In April 2022 Jürgen Sturm, Christian Kerl and Daniel Cremers were featured among the top 10 most influential scholars in robotics of the last decade.


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