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Locally isometric embeddings of quotients of the rotation group modulo finite symmetries

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  • Hielscher, Ralf
  • Lippert, Laura

Abstract

The analysis of manifold-valued data using embedding based methods is linked to the problem of finding suitable embeddings. In this paper we are interested in embeddings of quotient manifolds SO(3)∕S of the rotation group modulo finite symmetry groups. Data on such quotient manifolds naturally occur in crystallography, material science and biochemistry. We provide a generic framework for the construction of such embeddings which generalizes the embeddings constructed in Arnold et al. (2018). The central advantage of our larger class of embeddings is that it includes locally isometric embeddings for all crystallographic symmetry groups.

Suggested Citation

  • Hielscher, Ralf & Lippert, Laura, 2021. "Locally isometric embeddings of quotients of the rotation group modulo finite symmetries," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    • Handle: RePEc:eee:jmvana:v:185:y:2021:i:c:s0047259x21000427
    DOI: 10.1016/j.jmva.2021.104764
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    References listed on IDEAS

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    1. Pelletier, Bruno, 2005. "Kernel density estimation on Riemannian manifolds," Statistics & Probability Letters, Elsevier, vol. 73(3), pages 297-304, July.
    2. Arnold, R. & Jupp, P.E. & Schaeben, H., 2018. "Statistics of ambiguous rotations," Journal of Multivariate Analysis, Elsevier, vol. 165(C), pages 73-85.
    3. Hielscher, Ralf, 2013. "Kernel density estimation on the rotation group and its application to crystallographic texture analysis," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 119-143.
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