Construction of weighted crystallographic orientations capturing a given orientation density function
To be useful in numerical simulations of e.g. deformation processes, EBSD datasets of crystallographic orientations have to be downsized by several orders of magnitude yet preserving the orientation density function approximately. The objective is either to preserve the overall shape of the initiall...
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| Veröffentlicht in: | Journal of materials science Jg. 52; H. 4; S. 2077 - 2090 |
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| Abstract | To be useful in numerical simulations of e.g. deformation processes, EBSD datasets of crystallographic orientations have to be downsized by several orders of magnitude yet preserving the orientation density function approximately. The objective is either to preserve the overall shape of the initially kernel estimated orientation density function and in particular its non-negativity, or to preserve the unbiased estimates of the first Fourier coefficients up to a given finite order. Methods are presented how to construct a much smaller set of weighted orientations such that their kernel density estimate approximates the initial estimate. To preserve its overall shape the de la Vallée Poussin kernel is applied as it is the only known non-negative kernel with a finite Fourier series expansion avoiding truncation errors. If the first Fourier coefficients are to be preserved the Dirichlet kernel applies as it is the only kernel providing unbiased estimates of the Fourier coefficients up to any given finite order. The weights are determined numerically by resolving a least squares or a maximum likelihood problem. Due to the linearity of kernel density estimation and the Fourier transform the approaches in spatial and spectral domain are related to each other in a unique complementary way. For an exemplary practical application we use a large EBSD dataset of about 80.000 orientations from a recrystallized low alloyed Zirconium sheet. Our methods reduce the size of the dataset by about
99.75
%
to the order of 200 weighted orientations supporting a secondary approximate distribution with a volume portion of crystallites oriented differently than initially of less than
10
%
. |
|---|---|
| AbstractList | To be useful in numerical simulations of e.g. deformation processes, EBSD datasets of crystallographic orientations have to be downsized by several orders of magnitude yet preserving the orientation density function approximately. The objective is either to preserve the overall shape of the initially kernel estimated orientation density function and in particular its non-negativity, or to preserve the unbiased estimates of the first Fourier coefficients up to a given finite order. Methods are presented how to construct a much smaller set of weighted orientations such that their kernel density estimate approximates the initial estimate. To preserve its overall shape the de la Vallée Poussin kernel is applied as it is the only known non-negative kernel with a finite Fourier series expansion avoiding truncation errors. If the first Fourier coefficients are to be preserved the Dirichlet kernel applies as it is the only kernel providing unbiased estimates of the Fourier coefficients up to any given finite order. The weights are determined numerically by resolving a least squares or a maximum likelihood problem. Due to the linearity of kernel density estimation and the Fourier transform the approaches in spatial and spectral domain are related to each other in a unique complementary way. For an exemplary practical application we use a large EBSD dataset of about 80.000 orientations from a recrystallized low alloyed Zirconium sheet. Our methods reduce the size of the dataset by about [Formula omitted] to the order of 200 weighted orientations supporting a secondary approximate distribution with a volume portion of crystallites oriented differently than initially of less than [Formula omitted]. To be useful in numerical simulations of e.g. deformation processes, EBSD datasets of crystallographic orientations have to be downsized by several orders of magnitude yet preserving the orientation density function approximately. The objective is either to preserve the overall shape of the initially kernel estimated orientation density function and in particular its non-negativity, or to preserve the unbiased estimates of the first Fourier coefficients up to a given finite order. Methods are presented how to construct a much smaller set of weighted orientations such that their kernel density estimate approximates the initial estimate. To preserve its overall shape the de la Vallée Poussin kernel is applied as it is the only known non-negative kernel with a finite Fourier series expansion avoiding truncation errors. If the first Fourier coefficients are to be preserved the Dirichlet kernel applies as it is the only kernel providing unbiased estimates of the Fourier coefficients up to any given finite order. The weights are determined numerically by resolving a least squares or a maximum likelihood problem. Due to the linearity of kernel density estimation and the Fourier transform the approaches in spatial and spectral domain are related to each other in a unique complementary way. For an exemplary practical application we use a large EBSD dataset of about 80.000 orientations from a recrystallized low alloyed Zirconium sheet. Our methods reduce the size of the dataset by about [Formula: see text] to the order of 200 weighted orientations supporting a secondary approximate distribution with a volume portion of crystallites oriented differently than initially of less than [Formula: see text]. To be useful in numerical simulations of e.g. deformation processes, EBSD datasets of crystallographic orientations have to be downsized by several orders of magnitude yet preserving the orientation density function approximately. The objective is either to preserve the overall shape of the initially kernel estimated orientation density function and in particular its non-negativity, or to preserve the unbiased estimates of the first Fourier coefficients up to a given finite order. Methods are presented how to construct a much smaller set of weighted orientations such that their kernel density estimate approximates the initial estimate. To preserve its overall shape the de la Vallée Poussin kernel is applied as it is the only known non-negative kernel with a finite Fourier series expansion avoiding truncation errors. If the first Fourier coefficients are to be preserved the Dirichlet kernel applies as it is the only kernel providing unbiased estimates of the Fourier coefficients up to any given finite order. The weights are determined numerically by resolving a least squares or a maximum likelihood problem. Due to the linearity of kernel density estimation and the Fourier transform the approaches in spatial and spectral domain are related to each other in a unique complementary way. For an exemplary practical application we use a large EBSD dataset of about 80.000 orientations from a recrystallized low alloyed Zirconium sheet. Our methods reduce the size of the dataset by about \[99.75\,\%\] to the order of 200 weighted orientations supporting a secondary approximate distribution with a volume portion of crystallites oriented differently than initially of less than \[10\,\%\]. To be useful in numerical simulations of e.g. deformation processes, EBSD datasets of crystallographic orientations have to be downsized by several orders of magnitude yet preserving the orientation density function approximately. The objective is either to preserve the overall shape of the initially kernel estimated orientation density function and in particular its non-negativity, or to preserve the unbiased estimates of the first Fourier coefficients up to a given finite order. Methods are presented how to construct a much smaller set of weighted orientations such that their kernel density estimate approximates the initial estimate. To preserve its overall shape the de la Vallée Poussin kernel is applied as it is the only known non-negative kernel with a finite Fourier series expansion avoiding truncation errors. If the first Fourier coefficients are to be preserved the Dirichlet kernel applies as it is the only kernel providing unbiased estimates of the Fourier coefficients up to any given finite order. The weights are determined numerically by resolving a least squares or a maximum likelihood problem. Due to the linearity of kernel density estimation and the Fourier transform the approaches in spatial and spectral domain are related to each other in a unique complementary way. For an exemplary practical application we use a large EBSD dataset of about 80.000 orientations from a recrystallized low alloyed Zirconium sheet. Our methods reduce the size of the dataset by about 99.75 % to the order of 200 weighted orientations supporting a secondary approximate distribution with a volume portion of crystallites oriented differently than initially of less than 10 % . |
| Audience | Academic |
| Author | Fundenberger, Jean-Jacques Schaeben, Helmut Bachmann, Florian |
| Author_xml | – sequence: 1 givenname: Helmut orcidid: 0000-0002-0237-9857 surname: Schaeben fullname: Schaeben, Helmut email: schaeben@tu-freiberg.de organization: TU Bergakademie Freiberg – sequence: 2 givenname: Florian surname: Bachmann fullname: Bachmann, Florian organization: Xnovo Technology ApS – sequence: 3 givenname: Jean-Jacques surname: Fundenberger fullname: Fundenberger, Jean-Jacques organization: Université de Lorraine |
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| Cites_doi | 10.1093/biomet/74.4.751 10.1107/S0021889892009270 10.1109/TC.1976.1674577 10.1214/aoms/1177697523 10.1007/978-3-0348-7448-9 10.1016/S0022-5096(01)00016-3 10.1111/j.1365-2818.2007.01811.x 10.1214/aos/1176347628 10.2748/tmj/1178241489 10.1080/01621459.1996.10476701 10.1007/978-1-4899-3324-9 10.1007/978-1-4899-4493-1 10.1016/j.jmva.2013.03.014 10.2977/prims/1195194875 10.1016/j.commatsci.2007.09.015 10.1155/TSM.33.365 10.1016/j.jmatprotec.2006.10.006 10.4028/www.scientific.net/SSP.160.63 10.1007/b13794 10.1016/j.jmva.2005.03.009 10.1007/BF01972448 10.1002/1521-3951(199704)200:2<367::AID-PSSB367>3.0.CO;2-I 10.1093/biomet/71.2.353 10.1016/j.mechmat.2015.04.014 10.1016/j.spl.2005.04.004 10.1093/oso/9780198536826.001.0001 10.1002/9781118575574 10.2307/2291420 |
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| Keywords | Orientation Density Function (ODF) Poussin Kernel Kernel Density Estimate Dirichlet Kernel Electron Back Scatter Diffraction (EBSD) |
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