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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Schaeben</name><affiliation><mods:affiliation>LM2P/ISGMP, UFR MIM-Soiences, University of Metz, Ile du Saulcy, 57045, Metz-Cedex 01, France</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:434CE99B575325A4985D12F76C4F47E1191F73A6</idno><date when="1994" year="1994">1994</date><idno type="doi">10.1007/BF01578386</idno><idno type="url">https://api.istex.fr/document/434CE99B575325A4985D12F76C4F47E1191F73A6/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">001993</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">001993</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Analogy and duality of texture analysis by harmonics or indicators</title><author><name sortKey="Schaeben, H" sort="Schaeben, H" uniqKey="Schaeben H" first="H." last="Schaeben">H. Schaeben</name><affiliation><mods:affiliation>LM2P/ISGMP, UFR MIM-Soiences, University of Metz, Ile du Saulcy, 57045, Metz-Cedex 01, France</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">Journal of Scientific Computing</title><title level="j" type="abbrev">J Sci Comput</title><idno type="ISSN">0885-7474</idno><idno type="eISSN">1573-7691</idno><imprint><publisher>Kluwer Academic Publishers</publisher><pubPlace>Dordrecht</pubPlace><date type="published" when="1994-06-01">1994-06-01</date><biblScope unit="volume">9</biblScope><biblScope unit="issue">2</biblScope><biblScope unit="page" from="173">173</biblScope><biblScope unit="page" to="195">195</biblScope></imprint><idno type="ISSN">0885-7474</idno></series><idno type="istex">434CE99B575325A4985D12F76C4F47E1191F73A6</idno><idno type="DOI">10.1007/BF01578386</idno><idno type="ArticleID">BF01578386</idno><idno type="ArticleID">Art5</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0885-7474</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><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></front></TEI><istex><corpusName>springer</corpusName><author><json:item><name>H. Schaeben</name><affiliations><json:string>LM2P/ISGMP, UFR MIM-Soiences, University of Metz, Ile du Saulcy, 57045, Metz-Cedex 01, France</json:string></affiliations></json:item></author><articleId><json:string>BF01578386</json:string><json:string>Art5</json:string></articleId><language><json:string>eng</json:string></language><originalGenre><json:string>OriginalPaper</json:string></originalGenre><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.</abstract><qualityIndicators><score>8</score><pdfVersion>1.3</pdfVersion><pdfPageSize>432 x 648 pts</pdfPageSize><refBibsNative>false</refBibsNative><keywordCount>0</keywordCount><abstractCharCount>2802</abstractCharCount><pdfWordCount>6822</pdfWordCount><pdfCharCount>36252</pdfCharCount><pdfPageCount>23</pdfPageCount><abstractWordCount>424</abstractWordCount></qualityIndicators><title>Analogy and duality of texture analysis by harmonics or indicators</title><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. 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ICOTOM8</title><publicationDate>1988-09</publicationDate></host><title>Orientation distributions--Representation and determination</title><publicationDate>1988-09</publicationDate></json:item></refBibs><genre><json:string>research-article</json:string></genre><host><volume>9</volume><pages><last>195</last><first>173</first></pages><issn><json:string>0885-7474</json:string></issn><issue>2</issue><subject><json:item><value>Computational Mathematics and Numerical Analysis</value></json:item><json:item><value>Algorithms</value></json:item><json:item><value>Mathematical and Computational Physics</value></json:item><json:item><value>Appl.Mathematics/Computational Methods of Engineering</value></json:item></subject><journalId><json:string>10915</json:string></journalId><genre><json:string>journal</json:string></genre><language><json:string>unknown</json:string></language><eissn><json:string>1573-7691</json:string></eissn><title>Journal of Scientific Computing</title><publicationDate>1994</publicationDate><copyrightDate>1994</copyrightDate></host><categories><wos><json:string>science</json:string><json:string>mathematics, applied</json:string></wos><scienceMetrix><json:string>natural sciences</json:string><json:string>mathematics & statistics</json:string><json:string>applied mathematics</json:string></scienceMetrix></categories><publicationDate>1994</publicationDate><copyrightDate>1994</copyrightDate><doi><json:string>10.1007/BF01578386</json:string></doi><id>434CE99B575325A4985D12F76C4F47E1191F73A6</id><score>0.019364022</score><fulltext><json:item><extension>pdf</extension><original>true</original><mimetype>application/pdf</mimetype><uri>https://api.istex.fr/document/434CE99B575325A4985D12F76C4F47E1191F73A6/fulltext/pdf</uri></json:item><json:item><extension>zip</extension><original>false</original><mimetype>application/zip</mimetype><uri>https://api.istex.fr/document/434CE99B575325A4985D12F76C4F47E1191F73A6/fulltext/zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/434CE99B575325A4985D12F76C4F47E1191F73A6/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a" type="main" xml:lang="en">Analogy and duality of texture analysis by harmonics or indicators</title><respStmt><resp>Références bibliographiques récupérées via GROBID</resp><name resp="ISTEX-API">ISTEX-API (INIST-CNRS)</name></respStmt><respStmt><resp>Références bibliographiques récupérées via GROBID</resp><name resp="ISTEX-API">ISTEX-API (INIST-CNRS)</name></respStmt></titleStmt><publicationStmt><authority>ISTEX</authority><publisher>Kluwer Academic Publishers</publisher><pubPlace>Dordrecht</pubPlace><availability><p>Plenum Publishing Corporation, 1994</p></availability><date>1993-09-08</date></publicationStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a" type="main" xml:lang="en">Analogy and duality of texture analysis by harmonics or indicators</title><author xml:id="author-1"><persName><forename type="first">H.</forename><surname>Schaeben</surname></persName><affiliation>LM2P/ISGMP, UFR MIM-Soiences, University of Metz, Ile du Saulcy, 57045, Metz-Cedex 01, France</affiliation></author></analytic><monogr><title level="j">Journal of Scientific Computing</title><title level="j" type="abbrev">J Sci Comput</title><idno type="journal-ID">10915</idno><idno type="pISSN">0885-7474</idno><idno type="eISSN">1573-7691</idno><idno type="issue-article-count">7</idno><idno type="volume-issue-count">4</idno><imprint><publisher>Kluwer Academic Publishers</publisher><pubPlace>Dordrecht</pubPlace><date type="published" when="1994-06-01"></date><biblScope unit="volume">9</biblScope><biblScope unit="issue">2</biblScope><biblScope unit="page" from="173">173</biblScope><biblScope unit="page" to="195">195</biblScope></imprint></monogr><idno type="istex">434CE99B575325A4985D12F76C4F47E1191F73A6</idno><idno type="DOI">10.1007/BF01578386</idno><idno type="ArticleID">BF01578386</idno><idno type="ArticleID">Art5</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>1993-09-08</date></creation><langUsage><language ident="en">en</language></langUsage><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></abstract><textClass><keywords scheme="Journal Subject"><list><head>Mathematics</head><item><term>Computational Mathematics and Numerical Analysis</term></item><item><term>Algorithms</term></item><item><term>Mathematical and Computational Physics</term></item><item><term>Appl.Mathematics/Computational Methods of Engineering</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="1993-09-08">Created</change><change when="1994-06-01">Published</change><change xml:id="refBibs-istex" who="#ISTEX-API" when="2016-11-22">References added</change><change xml:id="refBibs-istex" who="#ISTEX-API" when="2017-01-20">References added</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/434CE99B575325A4985D12F76C4F47E1191F73A6/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="Springer, Publisher found" wicri:toSee="no header"><istex:xmlDeclaration>version="1.0" encoding="UTF-8"</istex:xmlDeclaration><istex:docType PUBLIC="-//Springer-Verlag//DTD A++ V2.4//EN" URI="http://devel.springer.de/A++/V2.4/DTD/A++V2.4.dtd" name="istex:docType"></istex:docType><istex:document><Publisher><PublisherInfo><PublisherName>Kluwer Academic Publishers</PublisherName><PublisherLocation>Dordrecht</PublisherLocation></PublisherInfo><Journal><JournalInfo JournalProductType="ArchiveJournal" NumberingStyle="Unnumbered"><JournalID>10915</JournalID><JournalPrintISSN>0885-7474</JournalPrintISSN><JournalElectronicISSN>1573-7691</JournalElectronicISSN><JournalTitle>Journal of Scientific Computing</JournalTitle><JournalAbbreviatedTitle>J Sci Comput</JournalAbbreviatedTitle><JournalSubjectGroup><JournalSubject Type="Primary">Mathematics</JournalSubject><JournalSubject Type="Secondary">Computational Mathematics and Numerical Analysis</JournalSubject><JournalSubject Type="Secondary">Algorithms</JournalSubject><JournalSubject Type="Secondary">Mathematical and Computational Physics</JournalSubject><JournalSubject Type="Secondary">Appl.Mathematics/Computational Methods of Engineering</JournalSubject></JournalSubjectGroup></JournalInfo><Volume><VolumeInfo VolumeType="Regular" TocLevels="0"><VolumeIDStart>9</VolumeIDStart><VolumeIDEnd>9</VolumeIDEnd><VolumeIssueCount>4</VolumeIssueCount></VolumeInfo><Issue IssueType="Regular"><IssueInfo TocLevels="0"><IssueIDStart>2</IssueIDStart><IssueIDEnd>2</IssueIDEnd><IssueArticleCount>7</IssueArticleCount><IssueHistory><CoverDate><Year>1994</Year><Month>6</Month></CoverDate></IssueHistory><IssueCopyright><CopyrightHolderName>Plenum Publishing Corporation</CopyrightHolderName><CopyrightYear>1994</CopyrightYear></IssueCopyright></IssueInfo><Article ID="Art5"><ArticleInfo Language="En" ArticleType="OriginalPaper" NumberingStyle="Unnumbered" TocLevels="0" ContainsESM="No"><ArticleID>BF01578386</ArticleID><ArticleDOI>10.1007/BF01578386</ArticleDOI><ArticleSequenceNumber>5</ArticleSequenceNumber><ArticleTitle Language="En">Analogy and duality of texture analysis by harmonics or indicators</ArticleTitle><ArticleFirstPage>173</ArticleFirstPage><ArticleLastPage>195</ArticleLastPage><ArticleHistory><RegistrationDate><Year>2005</Year><Month>4</Month><Day>12</Day></RegistrationDate><Received><Year>1993</Year><Month>9</Month><Day>8</Day></Received></ArticleHistory><ArticleCopyright><CopyrightHolderName>Plenum Publishing Corporation</CopyrightHolderName><CopyrightYear>1994</CopyrightYear></ArticleCopyright><ArticleGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ArticleGrants><ArticleContext><JournalID>10915</JournalID><VolumeIDStart>9</VolumeIDStart><VolumeIDEnd>9</VolumeIDEnd><IssueIDStart>2</IssueIDStart><IssueIDEnd>2</IssueIDEnd></ArticleContext></ArticleInfo><ArticleHeader><AuthorGroup><Author AffiliationIDS="Aff1" PresentAffiliationID="Aff2"><AuthorName DisplayOrder="Western"><GivenName>H.</GivenName><FamilyName>Schaeben</FamilyName></AuthorName></Author><Affiliation ID="Aff1"><OrgDivision>LM2P/ISGMP, UFR MIM-Soiences</OrgDivision><OrgName>University of Metz</OrgName><OrgAddress><Street>Ile du Saulcy</Street><Postcode>57045</Postcode><City>Metz-Cedex 01</City><Country>France</Country></OrgAddress></Affiliation><Affiliation ID="Aff2"><OrgDivision>Department of Geography/Geosciences</OrgDivision><OrgName>University of Trier</OrgName><OrgAddress><Postcode>54286</Postcode><City>Trier</City><Country>Germany</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>According to the classic harmonic approach, an orientation density function (odf)<Emphasis Type="Italic">f</Emphasis> is expanded into its corresponding Fourier orthogonal series with respect to generalized spherical harmonics, and a pole density function (pdf)<InlineEquation ID="IE1"><InlineMediaObject><ImageObject FileRef="10915_2005_Article_BF01578386_TeX2GIFIE1.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX"> $$\tilde P_h $$ </EquationSource></InlineEquation> 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></InlineMediaObject><EquationSource Format="TEX"> $$\tilde \tilde f$$ </EquationSource></InlineEquation> 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> is given, the even part<InlineEquation ID="IE3"><InlineMediaObject><ImageObject FileRef="10915_2005_Article_BF01578386_TeX2GIFIE3.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX"> $$\tilde f$$ </EquationSource></InlineEquation> can be mathematically represented explicitly in terms of<Emphasis Type="Italic">f</Emphasis> itself. Thus, it is possible to render maps of<Emphasis Type="Italic">harmonic orientation ghosts</Emphasis>, and to evaluate<Emphasis Type="Italic">variants of mathematical standard odfs</Emphasis> resulting in identical pdfs independent of pdf data. However, if only normal diffraction pdfs are known, the data-dependent<Emphasis Type="Italic">variation width of feasible odfs</Emphasis> 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> of orientations is expanded into its corresponding Fourier orthogonal series with respect to indicator functions of the elements of a partition of<Emphasis Type="Italic">G</Emphasis>, and a pdf<InlineEquation ID="IE4"><InlineMediaObject><ImageObject FileRef="10915_2005_Article_BF01578386_TeX2GIFIE4.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX"> $$\tilde P_h $$ </EquationSource></InlineEquation> defined on the upper (lower) unit hemisphere<Emphasis Type="Italic">S</Emphasis><Stack><Subscript>+</Subscript><Superscript>3</Superscript></Stack>⊂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></Stack>. 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>. Doing so for all Fourier coefficients in the finite series expansion, i.e. for all elements of the partition of the set<Emphasis Type="Italic">G</Emphasis>, eventually results in the data-dependent variation width of odfs feasible with respect to given pdf data, and to the partitions of<Emphasis Type="Italic">G</Emphasis> and<Emphasis Type="Italic">S</Emphasis><Stack><Subscript>+</Subscript><Superscript>3</Superscript></Stack>.</Para><Para>Thus it is confirmed that the appearance of orientation ghosts, in particular correlations of<Emphasis Type="Italic">true</Emphasis> and<Emphasis Type="Italic">ghost</Emphasis> 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<InlineEquation ID="IE5"><InlineMediaObject><ImageObject FileRef="10915_2005_Article_BF01578386_TeX2GIFIE5.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX"> $$\tilde f$$ </EquationSource></InlineEquation> can be determined uniquely and free of error is generally a reasonable initial condition to develop ghost correction procedures.</Para></Abstract><KeywordGroup Language="En"><Heading>Key Words</Heading><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></KeywordGroup></ArticleHeader><NoBody></NoBody></Article></Issue></Volume></Journal></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Analogy and duality of texture analysis by harmonics or indicators</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>Analogy and duality of texture analysis by harmonics or indicators</title></titleInfo><name type="personal"><namePart type="given">H.</namePart><namePart type="family">Schaeben</namePart><affiliation>LM2P/ISGMP, UFR MIM-Soiences, University of Metz, Ile du Saulcy, 57045, Metz-Cedex 01, France</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="OriginalPaper"></genre><originInfo><publisher>Kluwer Academic Publishers</publisher><place><placeTerm type="text">Dordrecht</placeTerm></place><dateCreated encoding="w3cdtf">1993-09-08</dateCreated><dateIssued encoding="w3cdtf">1994-06-01</dateIssued><copyrightDate encoding="w3cdtf">1994</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType></physicalDescription><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><relatedItem type="host"><titleInfo><title>Journal of Scientific Computing</title></titleInfo><titleInfo type="abbreviated"><title>J Sci Comput</title></titleInfo><genre type="journal" displayLabel="Archive Journal"></genre><originInfo><dateIssued encoding="w3cdtf">1994-06-01</dateIssued><copyrightDate encoding="w3cdtf">1994</copyrightDate></originInfo><subject><genre>Mathematics</genre><topic>Computational Mathematics and Numerical Analysis</topic><topic>Algorithms</topic><topic>Mathematical and Computational Physics</topic><topic>Appl.Mathematics/Computational Methods of Engineering</topic></subject><identifier type="ISSN">0885-7474</identifier><identifier type="eISSN">1573-7691</identifier><identifier type="JournalID">10915</identifier><identifier type="IssueArticleCount">7</identifier><identifier type="VolumeIssueCount">4</identifier><part><date>1994</date><detail type="volume"><number>9</number><caption>vol.</caption></detail><detail type="issue"><number>2</number><caption>no.</caption></detail><extent unit="pages"><start>173</start><end>195</end></extent></part><recordInfo><recordOrigin>Plenum Publishing Corporation, 1994</recordOrigin></recordInfo></relatedItem><identifier type="istex">434CE99B575325A4985D12F76C4F47E1191F73A6</identifier><identifier type="DOI">10.1007/BF01578386</identifier><identifier type="ArticleID">BF01578386</identifier><identifier type="ArticleID">Art5</identifier><accessCondition type="use and reproduction" contentType="copyright">Plenum Publishing Corporation, 1994</accessCondition><recordInfo><recordContentSource>SPRINGER</recordContentSource><recordOrigin>Plenum Publishing Corporation, 1994</recordOrigin></recordInfo></mods></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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