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Learning Rotations
  • Alberto Pepe,
  • Joan Lasenby,
  • Pablo Chacón
Alberto Pepe
University of Cambridge

Corresponding Author:[email protected]

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Joan Lasenby
University of Cambridge
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Pablo Chacón
Institute of Physical Chemistry Rocasolano
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Abstract

Many problems in computer vision today are solved via deep learning. Tasks like pose estimation from images, pose estimation from point clouds or structure from motion can all be formulated as a regression on rotations. However, there is no unique way of parametrizing rotations mathematically: matrices, quaternions, axis-angle representation or Euler angles are all commonly used in the field. Some of them, however, present intrinsic limitations, including discontinuities, gimbal lock or antipodal symmetry. These limitations may make the learning of rotations via neural networks a challenging problem, potentially introducing large errors. Following recent literature, we propose three case studies: a sanity check, a pose estimation from 3D point clouds and an inverse kinematic problem. We do so by employing a full geometric algebra (GA) description of rotations. We compare the GA formulation with a 6D continuous representation previously presented in the literature in terms of regression error and reconstruction accuracy. We empirically demonstrate that parametrizing rotations as bivectors outperforms the 6D representation. The GA approach overcomes the continuity issue of representations as the 6D representation does, but it also needs fewer parameters to be learned and offers an enhanced robustness to noise. GA hence provides a broader framework for describing rotations in a simple and compact way that is suitable for regression tasks via deep learning, showing high regression accuracy and good generalizability in realistic high-noise scenarios.
06 Dec 2021Submitted to Mathematical Methods in the Applied Sciences
07 Dec 2021Submission Checks Completed
07 Dec 2021Assigned to Editor
17 Dec 2021Reviewer(s) Assigned
20 Mar 2022Review(s) Completed, Editorial Evaluation Pending
02 May 2022Editorial Decision: Revise Major
25 May 20221st Revision Received
26 May 2022Submission Checks Completed
26 May 2022Assigned to Editor
28 May 2022Reviewer(s) Assigned
27 Jul 2022Review(s) Completed, Editorial Evaluation Pending
08 Aug 2022Editorial Decision: Revise Minor
18 Aug 20222nd Revision Received
19 Aug 2022Submission Checks Completed
19 Aug 2022Assigned to Editor
19 Aug 2022Review(s) Completed, Editorial Evaluation Pending
20 Aug 2022Editorial Decision: Accept
11 Sep 2022Published in Mathematical Methods in the Applied Sciences. 10.1002/mma.8698