Spectral Geometry Processing with Manifold Harmonics
Identifieur interne : 004733 ( Hal/Corpus ); précédent : 004732; suivant : 004734Spectral Geometry Processing with Manifold Harmonics
Auteurs : Bruno Vallet ; Bruno LévySource :
- Computer Graphics Forum [ 0167-7055 ] ; 2008.
Abstract
We present an explicit method to compute a generalization of the Fourier Transform on a mesh. It is well known that the eigenfunctions of the Laplace Beltrami operator (Manifold Harmonics) define a function basis allowing for such a transform. However, computing even just a few eigenvectors is out of reach for meshes with more than a few thousand vertices, and storing these eigenvectors is prohibitive for large meshes. To overcome these limitations, we propose a band-by-band spectrum computation algorithm and an out-of-core implementation that can compute thousands of eigenvectors for meshes with up to a million vertices. We also propose a limited-memory filtering algorithm, that does not need to store the eigenvectors. Using this latter algorithm, specific frequency bands can be filtered, without needing to compute the entire spectrum. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering. These technical achievements are supported by a solid yet simple theoretic framework based on Discrete Exterior Calculus (DEC). In particular, the issues of symmetry and discretization of the operator are considered with great care.
Url:
DOI: 10.1111/j.1467-8659.2008.01122.x
Links to Exploration step
Hal:inria-00331894Le document en format XML
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<front><div type="abstract" xml:lang="en">We present an explicit method to compute a generalization of the Fourier Transform on a mesh. It is well known that the eigenfunctions of the Laplace Beltrami operator (Manifold Harmonics) define a function basis allowing for such a transform. However, computing even just a few eigenvectors is out of reach for meshes with more than a few thousand vertices, and storing these eigenvectors is prohibitive for large meshes. To overcome these limitations, we propose a band-by-band spectrum computation algorithm and an out-of-core implementation that can compute thousands of eigenvectors for meshes with up to a million vertices. We also propose a limited-memory filtering algorithm, that does not need to store the eigenvectors. Using this latter algorithm, specific frequency bands can be filtered, without needing to compute the entire spectrum. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering. These technical achievements are supported by a solid yet simple theoretic framework based on Discrete Exterior Calculus (DEC). In particular, the issues of symmetry and discretization of the operator are considered with great care.</div>
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<idno type="stamp" n="INRIA-NANCY-GRAND-EST">INRIA Nancy - Grand Est</idno>
<idno type="stamp" n="LORIA">LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications</idno>
<idno type="stamp" n="LORIA-ACGI" p="LORIA">Algorithmique, calcul, image et géométrie</idno>
<idno type="stamp" n="INRIA2">INRIA 2</idno>
<idno type="stamp" n="INRIA-LORRAINE">INRIA Nancy - Grand Est</idno>
<idno type="stamp" n="LABO-LORIA-SET" p="LORIA">LABO-LORIA-SET</idno>
<idno type="stamp" n="UNIV-LORRAINE">Université de Lorraine</idno>
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<notesStmt><note type="audience" n="2">International</note>
<note type="popular" n="0">No</note>
<note type="peer" n="1">Yes</note>
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<sourceDesc><biblStruct><analytic><title xml:lang="en">Spectral Geometry Processing with Manifold Harmonics</title>
<author role="aut"><persName><forename type="first">Bruno</forename>
<surname>Vallet</surname>
</persName>
<idno type="halAuthorId">223874</idno>
<affiliation ref="#struct-3174"></affiliation>
</author>
<author role="aut"><persName><forename type="first">Bruno</forename>
<surname>Lévy</surname>
</persName>
<idno type="idHal">bruno-levy</idno>
<idno type="halAuthorId">136961</idno>
<idno type="arXiv">http://arxiv.org/a/brlevy</idno>
<affiliation ref="#struct-3174"></affiliation>
</author>
</analytic>
<monogr><idno type="halJournalId" status="VALID">12023</idno>
<idno type="issn">0167-7055</idno>
<idno type="eissn">1467-8659</idno>
<title level="j">Computer Graphics Forum</title>
<imprint><publisher>Wiley</publisher>
<biblScope unit="volume">27</biblScope>
<biblScope unit="issue">2</biblScope>
<biblScope unit="pp">251-260</biblScope>
<date type="datePub">2008</date>
</imprint>
</monogr>
<idno type="doi">10.1111/j.1467-8659.2008.01122.x</idno>
</biblStruct>
</sourceDesc>
<profileDesc><langUsage><language ident="en">English</language>
</langUsage>
<textClass><classCode scheme="acm" n="I.3"></classCode>
<classCode scheme="acm" n="I.3.5"></classCode>
<classCode scheme="halDomain" n="info.info-gr">Computer Science [cs]/Graphics [cs.GR]</classCode>
<classCode scheme="halTypology" n="ART">Journal articles</classCode>
</textClass>
<abstract xml:lang="en">We present an explicit method to compute a generalization of the Fourier Transform on a mesh. It is well known that the eigenfunctions of the Laplace Beltrami operator (Manifold Harmonics) define a function basis allowing for such a transform. However, computing even just a few eigenvectors is out of reach for meshes with more than a few thousand vertices, and storing these eigenvectors is prohibitive for large meshes. To overcome these limitations, we propose a band-by-band spectrum computation algorithm and an out-of-core implementation that can compute thousands of eigenvectors for meshes with up to a million vertices. We also propose a limited-memory filtering algorithm, that does not need to store the eigenvectors. Using this latter algorithm, specific frequency bands can be filtered, without needing to compute the entire spectrum. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering. These technical achievements are supported by a solid yet simple theoretic framework based on Discrete Exterior Calculus (DEC). In particular, the issues of symmetry and discretization of the operator are considered with great care.</abstract>
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