Analogy and duality of texture analysis by harmonics or indicators
Identifieur interne : 001993 ( Istex/Corpus ); précédent : 001992; suivant : 001994Analogy and duality of texture analysis by harmonics or indicators
Auteurs : H. SchaebenSource :
- Journal of Scientific Computing [ 0885-7474 ] ; 1994-06-01.
Abstract
Abstract: According to the classic harmonic approach, an orientation density function (odf)f is expanded into its corresponding Fourier orthogonal series with respect to generalized spherical harmonics, and a pole density function (pdf) $$\tilde P_h $$ into its corresponding Fourier orthogonal series with respect to spherical harmonics. While pdfs are even (antipodally symmetric) functions, odfs are generally not. Therefore, the part $$\tilde \tilde f$$ of the odf which cannot be determined from normal diffraction pdfs can be mathematically represented as the odd portion of its series expansion. If the odff is given, the even part $$\tilde f$$ can be mathematically represented explicitly in terms off itself. Thus, it is possible to render maps ofharmonic orientation ghosts, and to evaluatevariants of mathematical standard odfs resulting in identical pdfs independent of pdf data. However, if only normal diffraction pdfs are known, the data-dependentvariation width of feasible odfs remained unaccessible, and within the harmonic approach a measure of confidence in a solution of the pdf-to-odf inversion problem does not exist. According to any discrete approach, an odff defined on some setG of orientations is expanded into its corresponding Fourier orthogonal series with respect to indicator functions of the elements of a partition ofG, and a pdf $$\tilde P_h $$ defined on the upper (lower) unit hemisphereS + 3 ⊂ℝ3 into its corresponding Fourier orthogonal series with respect to indicator functions of the elements of a partition ofS + 3 . The ambiguity of the pdf-to-odf inversion problem is discussed in terms of column-rank deficiency of the augmented projection matrix. The implication of the harmonic approach to split an odf into auniquely determined and anundetermined part does no longer seem to be reasonable. However, it is possible to numerically determine data-dependent confidence intervals for the Fourier coefficients with respect to the indicator functions which can be immediately interpreted as mean orientation densities within the elements of the partition ofG. Doing so for all Fourier coefficients in the finite series expansion, i.e. for all elements of the partition of the setG, eventually results in the data-dependent variation width of odfs feasible with respect to given pdf data, and to the partitions ofG andS + 3 . Thus it is confirmed that the appearance of orientation ghosts, in particular correlations oftrue andghost orientation components, depends on the representation of an odf. It may be questioned whether in practical applications the implicit assumption of the harmonic method that the even part $$\tilde f$$ can be determined uniquely and free of error is generally a reasonable initial condition to develop ghost correction procedures.
Url:
DOI: 10.1007/BF01578386
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<front><div type="abstract" xml:lang="en">Abstract: According to the classic harmonic approach, an orientation density function (odf)f is expanded into its corresponding Fourier orthogonal series with respect to generalized spherical harmonics, and a pole density function (pdf) $$\tilde P_h $$ into its corresponding Fourier orthogonal series with respect to spherical harmonics. While pdfs are even (antipodally symmetric) functions, odfs are generally not. Therefore, the part $$\tilde \tilde f$$ of the odf which cannot be determined from normal diffraction pdfs can be mathematically represented as the odd portion of its series expansion. If the odff is given, the even part $$\tilde f$$ can be mathematically represented explicitly in terms off itself. Thus, it is possible to render maps ofharmonic orientation ghosts, and to evaluatevariants of mathematical standard odfs resulting in identical pdfs independent of pdf data. However, if only normal diffraction pdfs are known, the data-dependentvariation width of feasible odfs remained unaccessible, and within the harmonic approach a measure of confidence in a solution of the pdf-to-odf inversion problem does not exist. According to any discrete approach, an odff defined on some setG of orientations is expanded into its corresponding Fourier orthogonal series with respect to indicator functions of the elements of a partition ofG, and a pdf $$\tilde P_h $$ defined on the upper (lower) unit hemisphereS + 3 ⊂ℝ3 into its corresponding Fourier orthogonal series with respect to indicator functions of the elements of a partition ofS + 3 . The ambiguity of the pdf-to-odf inversion problem is discussed in terms of column-rank deficiency of the augmented projection matrix. The implication of the harmonic approach to split an odf into auniquely determined and anundetermined part does no longer seem to be reasonable. However, it is possible to numerically determine data-dependent confidence intervals for the Fourier coefficients with respect to the indicator functions which can be immediately interpreted as mean orientation densities within the elements of the partition ofG. Doing so for all Fourier coefficients in the finite series expansion, i.e. for all elements of the partition of the setG, eventually results in the data-dependent variation width of odfs feasible with respect to given pdf data, and to the partitions ofG andS + 3 . Thus it is confirmed that the appearance of orientation ghosts, in particular correlations oftrue andghost orientation components, depends on the representation of an odf. It may be questioned whether in practical applications the implicit assumption of the harmonic method that the even part $$\tilde f$$ can be determined uniquely and free of error is generally a reasonable initial condition to develop ghost correction procedures.</div>
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<refBibs><json:item><host><author><json:item><name>H,J Bunge</name>
</json:item>
</author>
<title>Mathematische Methoden der Texturanalyse</title>
<publicationDate>1969</publicationDate>
</host>
</json:item>
<json:item><author><json:item><name>H,J Bunge</name>
</json:item>
</author>
<host><author></author>
<title>trans. P. R. Morris) Texture Analysis in Materials Science</title>
<publicationDate>1982</publicationDate>
</host>
<publicationDate>1982</publicationDate>
</json:item>
<json:item><author><json:item><name>H,J Bunge</name>
</json:item>
<json:item><name>C Esling</name>
</json:item>
</author>
<host><volume>40</volume>
<pages><last>628</last>
<first>627</first>
</pages>
<author></author>
<title>J. Phys. Lett</title>
<publicationDate>1979</publicationDate>
</host>
<title>Determination of the odd part of the texture function</title>
<publicationDate>1979</publicationDate>
</json:item>
<json:item><host><author><json:item><name>P,L Butzer</name>
</json:item>
<json:item><name>R,J Nessel</name>
</json:item>
</author>
<title>Fourier Analysis and Approximation</title>
<publicationDate>1971</publicationDate>
</host>
</json:item>
<json:item><author><json:item><name>M Dahms</name>
</json:item>
<json:item><name>H,J Bunge</name>
</json:item>
</author>
<host><volume>10</volume>
<pages><last>35</last>
<first>21</first>
</pages>
<author></author>
<title>Textures and Microstructures</title>
<publicationDate>1988</publicationDate>
</host>
<title>A positivity method for the determination of complete orientation distribution functions</title>
<publicationDate>1988</publicationDate>
</json:item>
<json:item><author><json:item><name>C Esling</name>
</json:item>
<json:item><name>J Muller</name>
</json:item>
<json:item><name>H,J Bunge</name>
</json:item>
</author>
<host><volume>43</volume>
<pages><last>195</last>
<first>189</first>
</pages>
<author></author>
<title>J. Physique</title>
<publicationDate>1982</publicationDate>
</host>
<title>An integral formula for the even part of the texture function or The apparition of the fn and fa distributions</title>
<publicationDate>1982</publicationDate>
</json:item>
<json:item><author><json:item><name>A Haar</name>
</json:item>
</author>
<host><volume>69</volume>
<pages><last>371</last>
<first>331</first>
</pages>
<author></author>
<title>Math. Annalen</title>
<publicationDate>1910</publicationDate>
</host>
<title>Zur Theorie der orthogonalen Funktionensysteme</title>
<publicationDate>1910</publicationDate>
</json:item>
<json:item><author><json:item><name>E,T Jaynes</name>
</json:item>
</author>
<host><pages><last>952</last>
<first>939</first>
</pages>
<author></author>
<title>Proc. IEEE</title>
<publicationDate>1982</publicationDate>
</host>
<title>On the rationale of maximum-entropy methods</title>
<publicationDate>1982</publicationDate>
</json:item>
<json:item><author><json:item><name>A Lent</name>
</json:item>
</author>
<host><pages><last>257</last>
<first>249</first>
</pages>
<author></author>
<title>bnage Analysis and Evaluation, Society of Photographic Scientists and Engineers</title>
<publicationDate>1976</publicationDate>
</host>
<title>A convergent algorithm for maximum entropy image restoration, with a medical x-ray application</title>
<publicationDate>1976</publicationDate>
</json:item>
<json:item><author><json:item><name>A Lent</name>
</json:item>
<json:item><name>Y Censor</name>
</json:item>
</author>
<host><author></author>
<title>Mathemtical Programming, preprint</title>
<publicationDate>1989</publicationDate>
</host>
<title>The primal-dual algorithm as a constraint-set-manipulation device</title>
<publicationDate>1989</publicationDate>
</json:item>
<json:item><author><json:item><name>S Matthies</name>
</json:item>
</author>
<host><volume>92</volume>
<pages><last>138</last>
<first>135</first>
</pages>
<author></author>
<title>Phys. Stat. Sol. (b)</title>
<publicationDate>1979</publicationDate>
</host>
<title>On the reproducibility of the orientation distribution function of texture samples from pole figures (ghost phenomena)</title>
<publicationDate>1979</publicationDate>
</json:item>
<json:item><author><json:item><name>S Matthies</name>
</json:item>
</author>
<host><pages><last>48</last>
<first>37</first>
</pages>
<author></author>
<title>)0 Proe. 8th Int. Conf of Textures of Materials' (ICOTOMS)</title>
<publicationDate>1988-09</publicationDate>
</host>
<title>On the basic elements of and practical experiences with the WIMV algorithm-an odf reproduction method with conditional ghost correction</title>
<publicationDate>1988-09</publicationDate>
</json:item>
<json:item><author><json:item><name>S Matthies</name>
</json:item>
</author>
<host><pages><last>12</last>
<first>1</first>
</pages>
<author></author>
<title>Proc. 9th Inter. Conf. of Textures of Materials (ICOTOM9)</title>
<publicationDate>1990</publicationDate>
</host>
<title>On the principle of conditional ghost correction and its realization in existing correction concepts</title>
<publicationDate>1990</publicationDate>
</json:item>
<json:item><host><author><json:item><name>S Matthies</name>
</json:item>
<json:item><name>G,W Vinel</name>
</json:item>
<json:item><name>K Helming</name>
</json:item>
</author>
<title>Standard Distributions in Texture Analysis</title>
<publicationDate>1987</publicationDate>
</host>
</json:item>
<json:item><author><json:item><name>J Muller</name>
</json:item>
<json:item><name>C Esling</name>
</json:item>
<json:item><name>H,J Bunge</name>
</json:item>
</author>
<host><volume>42</volume>
<pages><last>165</last>
<first>161</first>
</pages>
<author></author>
<title>J. Physique</title>
<publicationDate>1981</publicationDate>
</host>
<title>An inversion formula expressing the texture function in terms of angular distribution functions</title>
<publicationDate>1981</publicationDate>
</json:item>
<json:item><author><json:item><name>B,A Murtagh</name>
</json:item>
<json:item><name>Saunders </name>
</json:item>
<json:item><name>M,A </name>
</json:item>
</author>
<host><volume>14</volume>
<pages><last>72</last>
<first>41</first>
</pages>
<author></author>
<title>Mathematical Programming</title>
<publicationDate>1978</publicationDate>
</host>
<title>Large-scale linearly constrained optimization</title>
<publicationDate>1978</publicationDate>
</json:item>
<json:item><author><json:item><name>D Ruer</name>
</json:item>
</author>
<host><author></author>
<title>These (Docteur es Sciences)</title>
<publicationDate>1976</publicationDate>
</host>
<title>Methode vectorielle d'analyse de la texture</title>
<publicationDate>1976</publicationDate>
</json:item>
<json:item><author><json:item><name>H Schaeben</name>
</json:item>
</author>
<host><volume>123</volume>
<pages><last>434</last>
<first>425</first>
</pages>
<author></author>
<title>Phys. Stat. SoL (b)</title>
<publicationDate>1984</publicationDate>
</host>
<title>Mathematical modeling of the orientation distribution function by the vector method of quantitative texture analysis</title>
<publicationDate>1984</publicationDate>
</json:item>
<json:item><author><json:item><name>H Schaeben</name>
</json:item>
</author>
<host><volume>148</volume>
<pages><last>72</last>
<first>63</first>
</pages>
<author></author>
<title>Methodology. Phys. Stat. Sol. (b)</title>
<publicationDate>1988</publicationDate>
</host>
<title>Entropy optimization in texture goniometry I</title>
<publicationDate>1988</publicationDate>
</json:item>
<json:item><author><json:item><name>H Schaeben</name>
</json:item>
</author>
<host><volume>158</volume>
<pages><last>425</last>
<first>407</first>
</pages>
<author></author>
<title>Phys. Stat. Sol. (b)</title>
<publicationDate>1990</publicationDate>
</host>
<title>Entropy optimization in texture geniometry II. Practical applications</title>
<publicationDate>1990</publicationDate>
</json:item>
<json:item><author><json:item><name>H Schaebem</name>
</json:item>
</author>
<host><volume>69</volume>
<pages><last>1329</last>
<first>1320</first>
</pages>
<author></author>
<title>J. Appl. Phys</title>
<publicationDate>1991</publicationDate>
</host>
<title>Entropy optimization in quantitative texture analysis, II Application to pole to orientation density inversion</title>
<publicationDate>1991</publicationDate>
</json:item>
<json:item><author><json:item><name>H Schaeben</name>
</json:item>
<json:item><name>J,J Fundenberger</name>
</json:item>
</author>
<host><volume>27</volume>
<pages><last>189</last>
<first>177</first>
</pages>
<author></author>
<title>J. Appl. Cryst</title>
<publicationDate>1994</publicationDate>
</host>
<title>Particular solutions of the pdf-to-odf inversion problem of texture analysis by large-scale mathemtical programming</title>
<publicationDate>1994</publicationDate>
</json:item>
<json:item><author><json:item><name>A Vadon</name>
</json:item>
<json:item><name>J,J Heizmann</name>
</json:item>
</author>
<host><pages><last>44</last>
<first>37</first>
</pages>
<author></author>
<title>Proc. 9th Int. Cot~ of Textures of Materials, (ICOTOM9 )</title>
<publicationDate>1990</publicationDate>
</host>
<title>A new program to calculate the texture vector for the vector method</title>
<publicationDate>1990</publicationDate>
</json:item>
<json:item><author><json:item><name>F Wagner</name>
</json:item>
<json:item><name>M Humbert</name>
</json:item>
<json:item><name>J Muller</name>
</json:item>
<json:item><name>C Esling</name>
</json:item>
</author>
<host><volume>11</volume>
<pages><last>483</last>
<first>479</first>
</pages>
<author></author>
<title>Europhys. Lett</title>
<publicationDate>1990</publicationDate>
</host>
<title>Optimization of the positivity method in quantitative texture analysis</title>
<publicationDate>1990</publicationDate>
</json:item>
<json:item><author><json:item><name>J,L Walsh</name>
</json:item>
</author>
<host><volume>45</volume>
<pages><last>24</last>
<first>5</first>
</pages>
<author></author>
<title>Am. J. Math</title>
<publicationDate>1923</publicationDate>
</host>
<title>A closed set of orthonormal functions</title>
<publicationDate>1923</publicationDate>
</json:item>
<json:item><host><author><json:item><name>H,R Wenk</name>
</json:item>
</author>
<title>Preferred Orientation #l Deformed Metals and Rocks--An Introduction to Modern Texture Analysis</title>
<publicationDate>1985</publicationDate>
</host>
</json:item>
<json:item><author><json:item><name>H,R Wenk</name>
</json:item>
<json:item><name>H,J Bunge</name>
</json:item>
<json:item><name>J,S Kallend</name>
</json:item>
<json:item><name>K Liicke</name>
</json:item>
<json:item><name>S Matthies</name>
</json:item>
<json:item><name>J Pospiech</name>
</json:item>
<json:item><name>Van Houtte</name>
</json:item>
<json:item><name>P </name>
</json:item>
</author>
<host><pages><last>30</last>
<first>17</first>
</pages>
<author></author>
<title>Proc. ICOTOM8</title>
<publicationDate>1988-09</publicationDate>
</host>
<title>Orientation distributions--Representation and determination</title>
<publicationDate>1988-09</publicationDate>
</json:item>
</refBibs>
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<surname>Schaeben</surname>
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<affiliation>LM2P/ISGMP, UFR MIM-Soiences, University of Metz, Ile du Saulcy, 57045, Metz-Cedex 01, France</affiliation>
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<monogr><title level="j">Journal of Scientific Computing</title>
<title level="j" type="abbrev">J Sci Comput</title>
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<abstract xml:lang="en"><p>Abstract: According to the classic harmonic approach, an orientation density function (odf)f is expanded into its corresponding Fourier orthogonal series with respect to generalized spherical harmonics, and a pole density function (pdf) $$\tilde P_h $$ into its corresponding Fourier orthogonal series with respect to spherical harmonics. While pdfs are even (antipodally symmetric) functions, odfs are generally not. Therefore, the part $$\tilde \tilde f$$ of the odf which cannot be determined from normal diffraction pdfs can be mathematically represented as the odd portion of its series expansion. If the odff is given, the even part $$\tilde f$$ can be mathematically represented explicitly in terms off itself. Thus, it is possible to render maps ofharmonic orientation ghosts, and to evaluatevariants of mathematical standard odfs resulting in identical pdfs independent of pdf data. However, if only normal diffraction pdfs are known, the data-dependentvariation width of feasible odfs remained unaccessible, and within the harmonic approach a measure of confidence in a solution of the pdf-to-odf inversion problem does not exist. According to any discrete approach, an odff defined on some setG of orientations is expanded into its corresponding Fourier orthogonal series with respect to indicator functions of the elements of a partition ofG, and a pdf $$\tilde P_h $$ defined on the upper (lower) unit hemisphereS + 3 ⊂ℝ3 into its corresponding Fourier orthogonal series with respect to indicator functions of the elements of a partition ofS + 3 . The ambiguity of the pdf-to-odf inversion problem is discussed in terms of column-rank deficiency of the augmented projection matrix. The implication of the harmonic approach to split an odf into auniquely determined and anundetermined part does no longer seem to be reasonable. However, it is possible to numerically determine data-dependent confidence intervals for the Fourier coefficients with respect to the indicator functions which can be immediately interpreted as mean orientation densities within the elements of the partition ofG. Doing so for all Fourier coefficients in the finite series expansion, i.e. for all elements of the partition of the setG, eventually results in the data-dependent variation width of odfs feasible with respect to given pdf data, and to the partitions ofG andS + 3 . Thus it is confirmed that the appearance of orientation ghosts, in particular correlations oftrue andghost orientation components, depends on the representation of an odf. It may be questioned whether in practical applications the implicit assumption of the harmonic method that the even part $$\tilde f$$ can be determined uniquely and free of error is generally a reasonable initial condition to develop ghost correction procedures.</p>
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<Para>According to the classic harmonic approach, an orientation density function (odf)<Emphasis Type="Italic">f</Emphasis>
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into its corresponding Fourier orthogonal series with respect to spherical harmonics. While pdfs are even (antipodally symmetric) functions, odfs are generally not. Therefore, the part<InlineEquation ID="IE2"><InlineMediaObject><ImageObject FileRef="10915_2005_Article_BF01578386_TeX2GIFIE2.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject>
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of the odf which cannot be determined from normal diffraction pdfs can be mathematically represented as the odd portion of its series expansion. If the odf<Emphasis Type="Italic">f</Emphasis>
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can be mathematically represented explicitly in terms of<Emphasis Type="Italic">f</Emphasis>
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remained unaccessible, and within the harmonic approach a measure of confidence in a solution of the pdf-to-odf inversion problem does not exist.</Para>
<Para>According to any discrete approach, an odf<Emphasis Type="Italic">f</Emphasis>
defined on some set<Emphasis Type="Italic">G</Emphasis>
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defined on the upper (lower) unit hemisphere<Emphasis Type="Italic">S</Emphasis>
<Stack><Subscript>+</Subscript>
<Superscript>3</Superscript>
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⊂ℝ<Superscript>3</Superscript>
into its corresponding Fourier orthogonal series with respect to indicator functions of the elements of a partition of<Emphasis Type="Italic">S</Emphasis>
<Stack><Subscript>+</Subscript>
<Superscript>3</Superscript>
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. The ambiguity of the pdf-to-odf inversion problem is discussed in terms of column-rank deficiency of the augmented projection matrix. The implication of the harmonic approach to split an odf into a<Emphasis Type="Italic">uniquely determined</Emphasis>
and an<Emphasis Type="Italic">undetermined</Emphasis>
part does no longer seem to be reasonable. However, it is possible to numerically determine data-dependent confidence intervals for the Fourier coefficients with respect to the indicator functions which can be immediately interpreted as mean orientation densities within the elements of the partition of<Emphasis Type="Italic">G</Emphasis>
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<Para>Thus it is confirmed that the appearance of orientation ghosts, in particular correlations of<Emphasis Type="Italic">true</Emphasis>
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can be determined uniquely and free of error is generally a reasonable initial condition to develop ghost correction procedures.</Para>
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<Keyword>Crystal orientation</Keyword>
<Keyword>tomographic inversion</Keyword>
<Keyword>texture analysis</Keyword>
<Keyword>generalized spherical harmonics</Keyword>
<Keyword>indicator functions</Keyword>
<Keyword>entropy optimization</Keyword>
<Keyword>large-scale programming</Keyword>
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<abstract lang="en">Abstract: According to the classic harmonic approach, an orientation density function (odf)f is expanded into its corresponding Fourier orthogonal series with respect to generalized spherical harmonics, and a pole density function (pdf) $$\tilde P_h $$ into its corresponding Fourier orthogonal series with respect to spherical harmonics. While pdfs are even (antipodally symmetric) functions, odfs are generally not. Therefore, the part $$\tilde \tilde f$$ of the odf which cannot be determined from normal diffraction pdfs can be mathematically represented as the odd portion of its series expansion. If the odff is given, the even part $$\tilde f$$ can be mathematically represented explicitly in terms off itself. Thus, it is possible to render maps ofharmonic orientation ghosts, and to evaluatevariants of mathematical standard odfs resulting in identical pdfs independent of pdf data. However, if only normal diffraction pdfs are known, the data-dependentvariation width of feasible odfs remained unaccessible, and within the harmonic approach a measure of confidence in a solution of the pdf-to-odf inversion problem does not exist. According to any discrete approach, an odff defined on some setG of orientations is expanded into its corresponding Fourier orthogonal series with respect to indicator functions of the elements of a partition ofG, and a pdf $$\tilde P_h $$ defined on the upper (lower) unit hemisphereS + 3 ⊂ℝ3 into its corresponding Fourier orthogonal series with respect to indicator functions of the elements of a partition ofS + 3 . The ambiguity of the pdf-to-odf inversion problem is discussed in terms of column-rank deficiency of the augmented projection matrix. The implication of the harmonic approach to split an odf into auniquely determined and anundetermined part does no longer seem to be reasonable. However, it is possible to numerically determine data-dependent confidence intervals for the Fourier coefficients with respect to the indicator functions which can be immediately interpreted as mean orientation densities within the elements of the partition ofG. Doing so for all Fourier coefficients in the finite series expansion, i.e. for all elements of the partition of the setG, eventually results in the data-dependent variation width of odfs feasible with respect to given pdf data, and to the partitions ofG andS + 3 . Thus it is confirmed that the appearance of orientation ghosts, in particular correlations oftrue andghost orientation components, depends on the representation of an odf. It may be questioned whether in practical applications the implicit assumption of the harmonic method that the even part $$\tilde f$$ can be determined uniquely and free of error is generally a reasonable initial condition to develop ghost correction procedures.</abstract>
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