Protected: 2018 CMU Workshop on Methods for 3D Microstructure Studies

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A Geodesic Octonion Metric for Grain Boundaries: Supplementary Material

This page provides supplementary material for an article that deals with the description of grain boundaries using octonions (8-D hypercomplex numbers).

T. Francis, I. Chesser, E.A. Holm and M. De Graef. "A New Octonion Metric for Grain Boundary Interpolations". Acta Materialia, 166:135-147 (2019). DOI:

 The main result of the paper is a geodesic arc length metric on the 7-sphere that expresses the distance metric between two grain boundaries that are characterized by quaternions with respect to a reference frame attached to the grain boundary.  This arc length can then be used in an interpolation formula to smoothly go from one grain boundary to a second one without ever leaving the 7-sphere.

The following links provide access to three different renderings corresponding to section 2 of the Supplementary Material:

The links below provide access to four different movies depicting the spherical linear interpolation or oSLERP between two octonions; each octonion represents particular grain boundary:

  • Interpolation between two symmetric tilt boundaries, with the grains represented by nested spheres: ​Movie 1​​​
  • Interpolation between two symmetric tilt boundaries, with the grains represented by cubes on either side of the grain boundary plane: ​Movie 2​​​
  • Interpolation between two random grain boundaries without application of crystal symmetry: ​Movie 3​​​
  • Interpolation between the same two random grain boundaries, this time with application of cubic symmetry which results in a different (shorter) geodesic pathway: ​Movie 4​​​

The complete Supplementary Materials document can be obtained from this link.

Iterative Pole Figure Inversion Approach

In this article, we propose a novel approach to the pole figure inversion problem, i.e., the reconstruction of the Orientation Distribution Function (ODF), based on a number of Pole Figures (PFs) derived either from X-ray diffraction measurements or from electron backscatter diffraction (EBSD) data.  Our method employs a modified equal-area Lambert projection for uniform sampling of S^2 as well as the equal volume "Cubochoric" representation for uniform discretization of orientation space, SO(3). The forward projection model is combined with penalty terms associated with lasso- and Tikhonov-based regularizations to iteratively determine the ODF. The efficacy of the new method is evaluated with both model and experimental data and compared to an existing method based on series expansions.

A draft version of the submitted manuscript can be found here.  The paper is currently under review with the Journal of Applied Crystallography.

The figures below represent the Supplementary Information that accompanies this paper.  These are renderings using the 3D stereographic projection representation as well as the homochoric (equal-volume) representation.  In each rendering, the ODF is represented as an emission cloud, the intensity corresponding  to the ODF value.  Crystallographic symmetry has been applied in all cases, meaning that for the cubic (octahedral) rotation group 432, the ODF section inside the fundamental zone (outlined in each rendering) is copied into 23 other equivalent components.  In the stereographic projections, the intensity of these clusters decreases with distance from the origin, since this is an equal-angle projection. In the homochoric representation, on the other hand, all clusters have the same intensity since this is an equal-volume representation.

3D Stereographic Projection animations in both regular and anaglyph mode for the three examples discussed in the paper. The Rodrigues Fundamental Zone is outlined in each animation.
Data Set Name Regular Anaglyph
Ground Truth Cube ODF spdf3-Cube-GroundTruth.mp4 spdf3-Cube-GroundTruth_anaglyph.mp4
- ADMM-l1 reconstruction spdf3-Cube-ADMM.mp4 spdf3-Cube-ADMM_anaglyph.mp4
- BCLS-l2 reconstruction spdf3-Cube-BCLS.mp4 spdf3-Cube-BCLS_anaglyph.mp4
Ground Truth Ti ODF spdf3-Ti-GroundTruth.mp4 spdf3-Ti-GroundTruth_anaglyph.mp4
- ADMM-l1 reconstruction spdf3-Ti-ADMM.mp4 spdf3-Ti-ADMM_anaglyph.mp4
- BCLS-l2 reconstruction spdf3-Ti-BCLS.mp4 spdf3-Ti-BCLS_anaglyph.mp4
Experimental Cu ODF
- ADMM-l1 reconstruction spdf3-Cu-ADMM.mp4 spdf3-Cu-ADMM_anaglyph.mp4
- BCLS-l2 reconstruction spdf3-Cu-BCLS.mp4 spdf3-Cu-BCLS_anaglyph.mp4