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

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