Statistical Estimation and Clustering of Group-invariant Orientation Parameters
This work addresses orientation estimation in materials science, specifically for crystal analysis, but is incremental as it builds on existing EM-ML frameworks with new parametric models.
The paper tackled the problem of estimating group-invariant orientation parameters, such as mean crystal orientations, by introducing hyperbolic Von Mises Fisher and Watson mixture models with a new EM-ML algorithm for clustering. Simulations and experiments on Electron Backscatter Diffraction data from a polycrystalline Nickel alloy sample demonstrated advantages of the extended estimators.
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.