Abstract
We describe a new algorithm to reconstruct a rigid body motion from point correspondences. The algorithm works by constructing a series of reflections which align the points with their correspondences one by one. This is naturally and efficiently implemented in the conformal model of geometric algebra, where the resulting transformation is represented by a versor. As a direct result of this algorithm, we also present a very compact and fast formula to compute a quaternion from two vector correspondences, a surprisingly elementary result which appears to be new.
This work was performed while the first author was at the University of Amsterdam
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- 1.
When applied within the conformal model, (4.2) can find pure rotations and pure translations, but no general rigid body motions which require a grade-4 part in their versor.
- 2.
An extension of this method to n-D (with n>6) may moreover be problematic since the Lagrangian constraints for the versor manifold are not yet known in general.
- 3.
All benchmarks were performed on a 2.8-GHz Intel Core2Duo processor using 64-bit (double) floating point precision.
- 4.
Editorial note: Chapter 7 gives a related method designed for redundant data.
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Acknowledgements
We acknowledge the support of NWO in the DASIS project (Discovery of Articulated Structures in Image Sequences) for funding this work. We are indebted to Richard Clawson who discovered and fixed the singularity in quaternion (4.5). For a more detailed description of the singularity, please refer to his write-up [3].
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Fontijne, D., Dorst, L. (2011). Reconstructing Rotations and Rigid Body Motions from Exact Point Correspondences Through Reflections. In: Dorst, L., Lasenby, J. (eds) Guide to Geometric Algebra in Practice. Springer, London. https://doi.org/10.1007/978-0-85729-811-9_4
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