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Quaternion: Difference between revisions
→Spatial Alignment Problem
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===Algebraic Solutions=== | ===Algebraic Solutions=== | ||
There are expressions for solving the eigenvalues of <math>M</math>. | There are expressions for solving the eigenvalues of <math>M</math>. | ||
==3D Orientation-Frame Alignment Problem== | |||
An orientation frame are the columns of the <math>SO(3)</math> orientation matrix. | |||
In this problem, we assume we have <math>n</math> paired orientation frames, <math>\{p_k\}</math> and <math>\{r_k\}</math> which we want to align using quaternion <math>q</math>. | |||
===Geodesic arc-length distance=== | |||
For two unit quaternions <math>q_1</math> and <math>q_2</math>, the geodesic arc length is <math>\alpha</math> where <math>q_1 \cdot q_2 = \cos \alpha</math>. | |||
Because of sign ambiguity (<math>R(q_2) = R(-q_2)</math>, we take the minimum value: | |||
<math display="block"> | |||
d_{geodesic}(q_1, q_2) = \min(\alpha, \pi-\alpha) \text{ where } | |||
0 \leq d_{geodesic}(q_1, q_2) \leq \frac{\pi}{2} | |||
</math> | |||
The least-squares optimization is: | |||
<math display="block"> | |||
\begin{align*} | |||
\mathbf{S}_{geodesic} &= \sum_{k=1}^{N}(\operatorname{arccos} |(q*p_k) \cdot r_k|)^2\\ | |||
&= \sum_{k=1}^{N}(\operatorname{arccos} |q \cdot (r_k * \bar{p}_k)|)^2 | |||
\end{align*} | |||
</math> | |||
==Resources== | ==Resources== | ||
* [https://arxiv.org/pdf/1804.03528.pdf The Quaternion-Based Spatial Coordinate and Orientation Frame Alignment Problems by Andrew Hanson] [https://journals.iucr.org/a/issues/2020/04/00/ib5072/ib5072.pdf Edited Paper] | * [https://arxiv.org/pdf/1804.03528.pdf The Quaternion-Based Spatial Coordinate and Orientation Frame Alignment Problems by Andrew Hanson] [https://journals.iucr.org/a/issues/2020/04/00/ib5072/ib5072.pdf Edited Paper] | ||