Symmetrized Bingham distribution for representing texture: parameter estimation with respect to crystal and sample symmetries

The quaternion Bingham distribution has been used to model preferred crystallographic orientation, or crystallographic texture, in polycrystalline materials in the materials science and geological communities. A primary difficulty in applying the Bingham distribution has been the lack of an efficient method for fitting the distribution parameters with respect to the material's underlying crystallographic symmetry or any statistical sample symmetry due to processing. This paper presents a symmetrized distribution, based on the quaternion Bingham, which can account for any general combination of crystallographic or sample symmetries. A numerical scheme is also introduced for estimating the parameters of the symmetrized distribution based on the well known expectation maximization algorithm.