  We treat the problem of estimation of orientation parameters whose values are
invariant to transformations from a spherical symmetry group. Previous work has
shown that any such group-invariant distribution must satisfy a restricted
finite mixture representation, which allows the orientation parameter to be
estimated using an Expectation Maximization (EM) maximum likelihood (ML)
estimation algorithm. In this paper, we introduce two parametric models for
this spherical symmetry group estimation problem: 1) the hyperbolic Von Mises
Fisher (VMF) mixture distribution and 2) the Watson mixture distribution. We
also introduce a new EM-ML algorithm for clustering samples that come from
mixtures of group-invariant distributions with different parameters. We apply
the models to the problem of mean crystal orientation estimation under the
spherically symmetric group associated with the crystal form, e.g., cubic or
octahedral or hexahedral. Simulations and experiments establish the advantages
of the extended EM-VMF and EM-Watson estimators for data acquired by Electron
Backscatter Diffraction (EBSD) microscopy of a polycrystalline Nickel alloy
sample.
