Parameter estimation in spherical symmetry groups
This provides a method for estimating mean crystal orientations in materials science, addressing a domain-specific problem with incremental improvements.
The paper tackles parameter estimation for probability distributions invariant under finite spherical symmetry groups by developing a group-invariant extension of the Von Mises Fisher distribution and an EM maximum likelihood estimator. Simulations and experiments on Electron Backscatter Diffraction data from a polycrystalline Nickel alloy show advantages of this estimator for mean crystal orientation estimation.
This paper considers statistical estimation problems where the probability distribution of the observed random variable is invariant with respect to actions of a finite topological group. It is shown that any such distribution must satisfy a restricted finite mixture representation. When specialized to the case of distributions over the sphere that are invariant to the actions of a finite spherical symmetry group $\mathcal G$, a group-invariant extension of the Von Mises Fisher (VMF) distribution is obtained. The $\mathcal G$-invariant VMF is parameterized by location and scale parameters that specify the distribution's mean orientation and its concentration about the mean, respectively. Using the restricted finite mixture representation these parameters can be estimated using an Expectation Maximization (EM) maximum likelihood (ML) estimation algorithm. This is illustrated for 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 VMF EM-ML estimator for data acquired by Electron Backscatter Diffraction (EBSD) microscopy of a polycrystalline Nickel alloy sample.