Parameter estimation in spherical symmetry groups

TL;DR

This paper introduces a group-invariant extension of the Von Mises Fisher distribution for spherical symmetry groups, enabling maximum likelihood estimation of mean orientation in crystal structures using EM algorithms, with demonstrated advantages in EBSD microscopy.

Contribution

It develops a novel group-invariant VMF distribution and an EM-based estimation method for symmetry-invariant distributions on the sphere.

Findings

The extended VMF distribution accurately models symmetry-invariant data.

The EM algorithm effectively estimates mean orientations in crystal structures.

Simulations and experiments show improved estimation performance.

Abstract

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.