Statistical estimation and clustering of group-invariant orientation parameters

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.

[1]  Alfred O. Hero,et al.  STATISTICAL METHODS FOR SIGNAL PROCESSING , 2005 .

[2]  M. Graef,et al.  Structure of Materials: An Introduction to Crystallography, Diffraction and Symmetry , 2004 .

[3]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[4]  S. R. Jammalamadaka,et al.  Directional Statistics, I , 2011 .

[5]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[6]  Alfred O. Hero,et al.  Coercive region-level registration for multi-modal images , 2015, 2015 IEEE International Conference on Image Processing (ICIP).

[7]  Alfred O. Hero,et al.  EBSD image segmentation using a physics-based forward model , 2013, 2013 IEEE International Conference on Image Processing.

[8]  Garrett Birkhoff,et al.  A brief survey of modern algebra , 1953 .

[9]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[10]  J. A. Hartigan,et al.  A k-means clustering algorithm , 1979 .

[11]  S. Altmann.  Rotations, Quaternions, and Double Groups , 1986 .

[12]  G. S. Watson,et al.  Equatorial distributions on a sphere , 1965 .

[13]  Knut Conradsen,et al.  Automated Determination of Crystal Orientations from Electron Backscattering Patterns , 1994 .

[14]  E. LESTER SMITH,et al.  AND OTHERS , 2005 .

[15]  Alfred O. Hero,et al.  Parameter Estimation in Spherical Symmetry Groups , 2015, IEEE Signal Processing Letters.

[16]  Hiromichi Nagasawa,et al.  Crystal orientation analyses of biominerals using Kikuchi patterns in TEM , 2007 .

[17]  S. S. Wilks.  The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses , 1938 .

[18]  Alfred O. Hero,et al.  Efficient learning of sparse, distributed, convolutional feature representations for object recognition , 2011, 2011 International Conference on Computer Vision.

[19]  Geoffrey J. McLachlan,et al.  Finite Mixture Models , 2019, Annual Review of Statistics and Its Application.