Unsupervised Learning for Efficient Texture Estimation From Limited Discrete Orientation Data
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THE connection between crystallographic preferred orientations (texture) in polycrystalline specimens and anisotropic material response/properties is widely recognized, and research into the quantitative analysis of texture has a long history in materials science and geology.[1,2,3,4,5,6,7] The relative frequency of crystal orientations within a sample is described mathematically via the orientation distribution function (ODF). Experimental techniques for the estimation of the ODF can be categorized as either macrotexture (or bulk/sample) texture techniques, such as X-ray or neutron diffraction,[8,9] where the relative frequency of orientations is averaged over hundreds of thousands of grains, or meso/ microtexture techniques, such as electron backscatter diffraction (EBSD)[10,11] or high-energy X-ray diffraction microscopy (HEDM),[12] which provide spatially resolved 2D or 3D orientation maps of a much smaller number of grains (hundreds to tens of thousands). Techniques for the estimation of orientation statistics from discrete orientation data grew organically out of the existing direct, such as the Williams–Imhof–Matthies–Vinel (WIMV) algorithm, and spherical harmonic STEPHEN R. NIEZGODA, Post-Doctoral Researcher, is with Materials Science in Radiation and Dynamic Extremes, Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM, 87545. Contact e-mail: [email protected] JARED GLOVER, Ph.D. Candidate, is with the Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA, 02139. Manuscript submitted December 31, 2012. METALLURGICAL AND MATERIALS TRANSACTIONS A
techniques for macrotexture analysis, and rely on building up the ODF from the superposition of contributions from the individual orientations.[1,3,13,14,15,16] The most common method for ODF estimation from discrete measurements is based on the generalized spherical harmonics championed by Bunge.[1] In the harmonic method, the Fourier coefficients for the ODF are estimated from the superposition of the Fourier coefficients of the individual orientations convolved with a smoothing kernel. This procedure relies heavily on assumptions made a priori on the type and degree of smoothing. In addition, ad-hoc methods are required both for choosing the bandwidth of the spherical harmonics as well as to enforce positivity constraints (to make a proper ODF). The degree of smoothing and bandwidth as well as the number of discrete orientations required for accurate ODF estimation has a long history of debate in the literature.[17,18,19,20] A recent advance, on this front, is the development of automatic smoothing kernel optimization available in the MTEX Quantitative Texture Analysis Software,[21] which assists the user in determining the appropriate level of smoothing and issues a warning if too low a bandwidth is used. Despite this advance, neither the direct nor harmonic approaches provide information-theoretic guarantees as to their optimality in describing a given dataset. As will be
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