Geometry

   

The Orthogonal Planes Split of Quaternions and Its Relation to Quaternion Geometry of Rotations

Authors: Eckhard Hitzer

Recently the general orthogonal planes split with respect to any two pure unit quaternions $f,g \in \mathbb{H}$, $f^2=g^2=-1$, including the case $f=g$, has proved extremely useful for the construction and geometric interpretation of general classes of double-kernel quaternion Fourier transformations (QFT) [E.Hitzer, S.J. Sangwine, The orthogonal 2D planes split of quaternions and steerable quaternion Fourier Transforms, in E. Hitzer, S.J. Sangwine (eds.), "Quaternion and Clifford Fourier Transforms and Wavelets", TIM \textbf{27}, Birkhauser, Basel, 2013, 15--39.]. Applications include color image processing, where the orthogonal planes split with $f=g=$ the grayline, naturally splits a pure quaternionic three-dimensional color signal into luminance and chrominance components. Yet it is found independently in the quaternion geometry of rotations [L. Meister, H. Schaeben, A concise quaternon geometry of rotations, MMAS 2005; \textbf{28}: 101--126], that the pure quaternion units $f,g$ and the analysis planes, which they define, play a key role in the spherical geometry of rotations, and the geometrical interpretation of integrals related to the spherical Radon transform of probability density functions of unit quaternions, as relevant for texture analysis in crystallography. In our contribution we further investigate these connections.

Comments: 10 Pages. Submitted to Proceedings of the 30th International Colloquium on Group Theoretical Methods in Physics (troup30), 14-18 July 2014, Ghent, Belgium, to be published by IOP in the Journal of Physics: Conference Series (JPCS), 2014.

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[v1] 2014-11-19 03:58:28

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