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Particular solutions of the PDF‐to‐ODF inversion problem of texture analysis by large‐scale mathematical programming
Author(s) -
Schaeben H.,
Fundenberger J. J.
Publication year - 1994
Publication title -
journal of applied crystallography
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.429
H-Index - 162
ISSN - 1600-5767
DOI - 10.1107/s0021889893007782
Subject(s) - mathematics , discretization , orientation (vector space) , upper and lower bounds , mathematical analysis , geometry
Discretizing the mathematically ill‐posed inversion problem of texture analysis to calculate an orientation density defined on G ⊂ SO (3) from given pole densities defined on S 3 + ⊂ 3 results in a largely column rank deficient system of linear equations. Particular model solutions of this system which minimize or maximize, respectively, given objective functions like texture index or entropy are calculated and compared for the Sante Fe texture problem. It is shown that maximization of the texture index is generally not a device for conditional ghost correction. Furthermore, particular solutions with extreme values for a given texture component g 0 ∈ G n 0 ∈ ( G n ) n = 1,⋯ N of the partition of the (Euler) orientation space G are calculated and discussed. Numerical evidence is given for the conjecture that application of a positive lower bound 0 < b to all orientation density functions (ODFs) f ∈ that are feasible with respect to a given set of pole density functions (PDFs), i.e. restriction to feasible ODFs 0 < b ≤ f ( g ), g ∈ G , f ∈ with a given minimum orientation density or uniform portion, respectively, substantially diminishes their variation. Finally, it is shown that this approach of optimizing individual texture components generalizes to the numerical determination of the variation width of feasible ODFs with respect to a given set of PDFs and the discretization of the inversion problem.