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Recasting Classical Expansion of orientation distribution function as Tensorial Fourier Expansion

JOURNAL OF ELASTICITY 2024年第1-5期155卷 109-131页

作者： Man, Chi-Sing Du, Wenwen Univ Kentucky Dept Math Lexington KY 40506 USA Glenville State Univ Dept Sci & Math Glenville WV 26351 USA

In classical texture analysis the orientation distribution function (ODF) is expanded as an infinite series of Wigner D -functions D-mn(l) (.), with the classical texture coefficients c(mn)(l) serving as undetermined expansion parameters. Herein we present a systematic procedure by which the classical expansion of an ODF (with arbitrary crystallite and texture symmetry) truncated at the order / = L can be directly rewritten as a tensorial Fourier expansion truncated at the same order, where the same set of classical texture coefficients c(mn)(l) appear as undetermined parameters. The crucial step of this procedure is as follows: Let H-1 be the complexified space of l-th order harmonic tensors and .,. the Hermitian inner product onz. Let R be a rotation and R-circle times l : H-1 -> H(1)be the l-th order tensor product of R with itself. For each 1 <= 1 <= L, determine an orthonormal irreducible tensor basis B-m(l) (-1 <= m <= 1) in H-1 so that D-mn(l) (R) = B-m(l), R-circle times l B-n(l)) for each rotation R. We have determined explicit sets of irreducible basis tensors B-m(l) in H-1 for 1 <= 1 <= 8, which are displayed in tables in this paper. In addition to examples which illustrate the general procedure, we provide, for orthorhombic aggregates of cubic and hexagonal crystallites, explicit expressions of tensorial Fourier expansions of the ODF up to terms with 1 = 8, in which the tensorial texture coefficients are given in terms of the classical texture coefficients and the basis tensors B-m(l) .

关键词： Tensorial texture coefficients Tensorial Fourier expansion Irreducible tensor basis Crystallographic texture orientation distribution function

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Reconstruction of the orientation distribution function for Materials with Low Lattice and Sample Symmetry Using the Harmonic Method

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INORGANIC MATERIALS 2024年第2期60卷 105-111页

作者： Kolyanova, A. S. Russian Acad Sci IMET RAS Baikov Inst Met & Mat Sci Moscow 119334 Russia

Many properties of polycrystalline materials depend on their crystallographic texture, which canbe fully described by the orientation distribution function (ODF). The main task of quantitative texture anal-ysis is to reconstruct the ODF from its two-dimensional projections, which are obtained through X-ray orneutron diffraction methods. In this work, the results of ODF reconstruction for materials with low latticeand sample symmetry using the harmonic method are presented. The method is based on expanding theODF in a Fourier series using three-dimensional symmetric spherical functions. Real functions which arelinear combinations of corresponding complex spherical functions were used. A model single-componenttexture and the texture of a magnesium alloy sample subjected to equal-channel angular pressing were inves-tigated. Both textures exhibit hexagonal lattice symmetry and triclinic sample symmetry. In both cases, theRP-factor values and the error of ODF calculation, used to check the adequacy of the solution, showed goodagreement between the calculated and original data. It was also found that the ODF of the magnesium alloysample contains two texture components ( 2 6)[ 211] and ( 2 6)[ 2 1] with maximum intensities of 13.81and 2.23, respectively. The obtained results can be used in texture studies of ceramics, rocks, and other non-metallic materials with low symmetry.

关键词： orientation distribution function harmonic method Fourier coefficients spherical harmonics low symmetry

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Performance of orientation distribution function-fingerprinting with a biophysical multicompartment diffusion model

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MAGNETIC RESONANCE IN MEDICINE 2022年第1期88卷 418-435页

作者： Filipiak, Patryk Shepherd, Timothy Lin, Ying-Chia Placantonakis, Dimitris G. Boada, Fernando E. Baete, Steven H. NYU Langone Hlth Ctr Adv Imaging Innovat & Res CAI2R Dept Radiol New York NY USA NYU Langone Hlth Dept Neurosurg Perlmutter Canc Ctr Neurosci InstKimmel Ctr Stem Cell Biol New York NY USA Stanford Univ Radiol Sci Lab Stanford CA 94305 USA Stanford Univ Mol Imaging Program Stanford Dept Radiol Stanford CA 94305 USA

Purpose: orientation distribution function (ODF) peak finding methods typically fail to reconstruct fibers crossing at shallow angles below 40., leading to errors in tractography. ODF-Fingerprinting (ODF-FP) with the biophysical multicompartment diffusion model allows for breaking this barrier. Methods: A randomized mechanism to generate a multidimensional ODF-dictionary that covers biologically plausible ranges of intra- and extra-axonal diffusivities and fraction volumes is introduced. This enables ODF-FP to address the high variability of brain tissue. The performance of the proposed approach is evaluated on both numerical simulations and a reconstruction of major fascicles from high- and low-resolution in vivo diffusion images. Results: ODF-FP with the suggested modifications correctly identifies fibers crossing at angles as shallow as 10 degrees in the simulated data. In vivo, our approach reaches 56% of true positives in determining fiber directions, resulting in visibly more accurate reconstruction of pyramidal tracts, arcuate fasciculus, and optic radiations than the state-of-the-art techniques. Moreover, the estimated diffusivity values and fraction volumes in corpus callosum conform with the values reported in the literature. Conclusion: The modified ODF-FP outperforms commonly used fiber reconstruction methods at shallow angles, which improves deterministic tractography outcomes of major fascicles. In addition, the proposed approach allows for linearization of the microstructure parameters fitting problem.

关键词： crossing fibers diffusion MRI fingerprinting microstructure model fitting multicompartment diffusion model orientation distribution function shallow angles tractography

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Performance of orientation distribution function-fingerprinting with a biophysical multicompartment diffusion model

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Magnetic Resonance in Medicine 2022年第1期88卷 C1页

作者： Patryk Filipiak Timothy Shepherd Ying-Chia Lin Dimitris G. Placantonakis Fernando E. Boada Steven H. Baete Center for Advanced Imaging Innovation and Research (CAI2R) Department of Radiology NYU Langone Health New York New York USA Department of Neurosurgery Perlmutter Cancer Center Neuroscience Institute Kimmel Center for Stem Cell Biology NYU Langone Health New York New York USA Radiological Sciences Laboratory and Molecular Imaging Program at Stanford Department of Radiology Stanford University Stanford California USA

Purpose orientation distribution function (ODF) peak finding methods typically fail to reconstruct fibers crossing at shallow angles below 40°, leading to errors in tractography. ODF-Fingerprinting (ODF-FP) with the biophysical multicompartment diffusion model allows for breaking this barrier. Methods A randomized mechanism to generate a multidimensional ODF-dictionary that covers biologically plausible ranges of intra- and extra-axonal diffusivities and fraction volumes is introduced. This enables ODF-FP to address the high variability of brain tissue. The performance of the proposed approach is evaluated on both numerical simulations and a reconstruction of major fascicles from high- and low-resolution in vivo diffusion images. Results ODF-FP with the suggested modifications correctly identifies fibers crossing at angles as shallow as 10 degrees in the simulated data. In vivo, our approach reaches 56% of true positives in determining fiber directions, resulting in visibly more accurate reconstruction of pyramidal tracts, arcuate fasciculus, and optic radiations than the state-of-the-art techniques. Moreover, the estimated diffusivity values and fraction volumes in corpus callosum conform with the values reported in the literature. Conclusion The modified ODF-FP outperforms commonly used fiber reconstruction methods at shallow angles, which improves deterministic tractography outcomes of major fascicles. In addition, the proposed approach allows for linearization of the microstructure parameters fitting problem.

关键词： crossing fibers diffusion MRI fingerprinting microstructure model fitting multicompartment diffusion model orientation distribution function shallow angles tractography

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A multivariate grain size and orientation distribution function: derivation from electron backscatter diffraction data and applications

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JOURNAL OF APPLIED CRYSTALLOGRAPHY 2021年第1期54卷 148-162页

作者： Lopez, Jesus Galan Kestens, Leo A., I Delft Univ Technol Fac 3mE MSE Mekelweg 2 NL-2628 CD Delft Netherlands Univ Ghent Fac Engn & Architecture Dept Electromech Syst & Met Engn Technol Pk Zwijnaarde 903 B-9052 Ghent Belgium

Two of the microstructural parameters most influential in the properties of polycrystalline materials are grain size and crystallographic texture. Although both properties have been extensively studied and there are a wide range of analysis tools available, they are generally considered independently, without taking into account the possible correlations between them. However, there are reasons to assume that grain size and orientation are correlated microstructural state variables, as they are the result of single microstructural formation mechanisms occurring during material processing. In this work, the grain size distribution and orientation distribution functions are combined in a single multivariate grain size orientation distribution function (GSODF). In addition to the derivation of the function, several examples of practical applications to low carbon steels are presented, in which it is shown how the GSODF can be used in the analysis of 2D and 3D electron backscatter diffraction data, as well as in the generation of representative volume elements for full-field models and as input in simulations using mean-field methods.

关键词： grain size texture orientation distribution function ODF electron backscatter diffraction EBSD representative volume element RVE

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orientation distribution function Pattern for Rigid Dumbbell Suspensions in Any Simple Shear Flow

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MACROMOLECULAR THEORY AND SIMULATIONS 2019年第1期28卷 1800046-1800046页

作者： Jbara, Layal M. Giacomin, Alan Jeffrey Queens Univ Chem Engn Dept Kingston ON K7L 3N6 Canada Queens Univ Polymers Res Grp Kingston ON K7L 3N6 Canada Queens Univ Mech & Mat Engn Dept Kingston ON K7L 3N6 Canada

For rigid dumbbells suspended in a Newtonian solvent, the viscoelastic response depends exclusively on the dynamics of dumbbell orientation. The orientation distribution function psi(theta, phi, t) represents the probability of finding dumbbells within the range (theta, theta + d theta) and (phi, phi + d phi). This function is expressed in terms of a partial differential equation (the diffusion equation), which, for any simple shear flow, is solved by postulating a series expansion in the shear rate magnitude. Each order of this expansion yields a new partial differential equation, for which one must postulate a form for its solution. This paper finds a simple and direct pattern to these solutions. The use of this pattern reduces the amount of work required to determine the coefficients of the power series expansion of the orientation distribution function, psi(i). To demonstrate the usefulness of this new pattern, new expressions for these coefficients are derived up to and including the sixth power of the shear rate magnitude. This work also completes previous findings that ended at the fourth power of the shear rate magnitude.

关键词： diffusion equation orientation distribution function rigid dumbbell simple shear flow

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Restoration of orientation distribution function Using Texture Components with Radial Normal distributions

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INORGANIC MATERIALS 2018年第15期54卷 1477-1482页

作者： Ivanova, T. M. Serebryanyi, V. N. Natl Res Nucl Univ MEPhI Moscow Russia Russian Acad Sci Baikov Inst Met & Mat Sci Moscow 119334 Russia

The methods for restoration of the orientation distribution function (ODF) from experimental pole figures have been compared for materials possessing a low symmetry of specimen (by example of the median section of the hot-pressed band from Mg-4.5% Nd magnesium alloy), namely, the method of texture components using radial normal distributions on SO(3) with different spreading and the method of superposition of a large number of normal distributions with equivalent small spreading. Both approaches have demonstrated similar ODFs. In this case, the former method, which is less sensitive to measurement errors of pole figures, is based on nonlinear optimization with a complex choice of initial approximations of the parameters of the model. The latter approach is more sensitive and easier to use.

关键词： orientation distribution function method of texture components method of superposition of a large number of radial normal distributions pole figure Mg-4 5% Nd magnesium alloy

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Simultaneous Prediction of Bendability and Deep Drawability Using orientation distribution function for Aluminum Alloy Sheets

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MATERIALS TRANSACTIONS 2018年第3期59卷 399-405页

作者： Inoue, Hirofumi Osaka Prefecture Univ Grad Sch Engn Dept Mat Sci Sakai Osaka 5998531 Japan

Sheet metal formability is generally affected by crystallographic texture. In particular, bendability and deep-drawability of aluminum and its alloys are closely related to the recrystallization texture of the rolled sheets. It is necessary to quantitatively predict them from a viewpoint of texture control. This paper described a method for simultaneous prediction of both the bendability and the deep-drawability on the basis of the average Taylor factor as a polycrystal calculated by using an orientation distribution function. The normalized Taylor factor (M-n-value) and the r-value were used as measures of bendability and deep-drawability, respectively. The predicted results from ideal orientations demonstrated that {001}<100> orientation had excellent bendability and poor deep-drawability, whereas {111}<110> orientation had poor bendability and excellent deep-drawability. The predicted results for some aluminum alloys suggested that conventional cold-rolled and annealed sheets would be favorable to bendability, and the addition of asymmetric warm rolling after cold rolling would lead to improved deep-drawability.

关键词： aluminum alloy sheet metal formability orientation distribution function average Taylor factor r-value

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Parcellation of Human Amygdala Subfields Using orientation distribution function and Spectral K-means Clustering 1

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MICCAI Workshop on Computational Diffusion MRI

作者： Wen, Qiuting Stirling, Brian D. Sha, Long Shen, Li Whalen, Paul J. Wu, Yu-Chien Indiana Univ Sch Med Dept Radiol & Imaging Sci Ctr Neuroimaging Goodman Hall355 West 16th StSuite 4100 Indianapolis IN 46202 USA Univ Southern Calif Div Biokinesiol & Phys Therapy Los Angeles CA 90089 USA Dartmouth Coll Dept Psychol & Brain Sci 6207 Moore Hall Hanover NH 03755 USA Dartmouth Coll Dartmouth Brain Imaging Ctr 6207 Moore Hall Hanover NH 03755 USA NYU Neurosci Inst 550 1St Ave New York NY 10016 USA

ISBN: (数字)9783319541303

ISBN: (纸本)9783319541303;9783319541297

Amygdala plays an important role in fear and emotional learning, which are critical for human survival. Despite the functional relevance and unique circuitry of each human amygdaloid subnuclei, there has yet to be an efficient imaging method for identifying these regions in vivo. A data-driven approach without prior knowledge provides advantages of efficient and objective assessments. The present study uses high angular and high spatial resolution diffusion magnetic resonance imaging to generate orientation distribution function, which bears distinctive microstructural features. The features were extracted using spherical harmonic decomposition to assess microstructural similarity within amygdala subfields that are identified via similarity matrices using spectral k-mean clustering. The approach was tested on 32 healthy volunteers and three distinct amygdala subfields were identified including medial, posterior-superior lateral, and anterior-inferior lateral.

关键词： distribution functions Angular distribution Clustering algorithms Diffusion Magnetic resonance imaging Microstructural evolution Data driven approach Diffusion magnetic resonance imaging functional relevance High spatial resolution K means clustering Microstructural features Objective assessment orientation distribution function

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The effect of approximation accuracy of the orientation distribution function on the elastic properties of short fibre reinforced composites

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COMPOSITE STRUCTURES 2016年 148卷 12-18页

作者： Eik, Marika Puttonen, Jari Herrmann, Heiko Aalto Univ Dept Civil Engn Sch Engn Rakentajanaukio 4 A Espoo Finland Tallinn Univ Technol Ctr Nonlinear Studies Inst Cybernet Akad Tee 21 EE-12618 Tallinn Estonia

In present paper the authors are focusing on the features of the orientation distribution function (ODF), used to give a probabilistic assessment to the orientation of fibres in fibre reinforced composites. The ODF can be approximated using experimental data and used to evaluate the anisotropic effect of fibres in considered directions. The hardened orthotropic elastic properties of steel fibre reinforced concrete (SFRC), evaluated based on the approximated ODF with different orders of accuracy are used as a case study. Input data for the approximation of the ODF are individual orientations of fibres measured by X-ray micro-tomography method. The study outcomes revealed that the orthotropic elastic properties of SFRC may not be estimated accurately enough by using only the second order terms in the reconstruction of the ODF. The accuracy of the stiffness matrix components is sensitive for the orientation distribution type and the approximation of the ODF, and this should be considered in all FE applications using the ODF approach. Besides, within the study the relation between the values of the scalar-order parameter S, evaluating how well the fibres are aligned with each other, and the approximation orders of the ODF is established. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： orientation distribution function Approximation accuracy Steel fibre reinforced concrete Orthotropic elasticity

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