Mesh Parameterization: Theory and Practice Global Methods

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Mesh ParameterizationTheory and
PracticeGlobal Methods
Bruno Lévy - INRIA
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Overview

• 1. Introduction - Motivations - Difficulties
• 2. Base Complex
• 3. Exterior Calculus
• 4. Spectral methods
• Epilogue / Conclusion

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1. Motivations
Generate a "coordinate system"
Texture atlas
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1. MotivationsArbitrary Topology Need for new
methods
Create a  geographic coordinate system 
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1. MotivationsRemeshing and Geometry Images
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1. MotivationsReverse engineering
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1. DifficultiesSingularities
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1. DifficultiesSingularities
Archibald Higgins Adventures (used with author
permission) For french readers, Archibald is
known as Anselme Lanturlu http//www.savoir-sans
-frontieres.org
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1. DifficultiesSingularities
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1. DifficultiesSingularities
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1. DifficultiesSingularities
Poincarré Hopf
DEC version N-Symmetry direction fields Ray
et.al See also N-Rosy Zhang et.al
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1. DifficultiesFunctions over manifolds
Differential Manifold see, e.g. Grimm et.al,
Gortler et.al, Ray et.al
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2. Base ComplexTriangulated Base Complex
Multires Analysis, MAPS, Inter-Chart Relaxation
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2. Base ComplexQuadrangulated Base Complex
Polycubemaps
Semi-manual construction
Tarini et.al
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2. Base ComplexWhy quads are desirable ?
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2. Base ComplexFrustration with charts
Lots of computations ! Handling charts is a hassle
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3. Exterior Calculus to the rescue ...

• Invented by Elie Cartan
• Pinkall Polthier
• Gu Yau
• Desbrun Schroeder, DEC, DDG
• Joy of EC (Google search it)
• Wikipedia
• Course notes

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3. Exterior Calculus

• Manipulate functions and their derivatives on
  manifolds directly
• Basic theorem generalizes Stokes theorem (flux
  through closed loop)
• Integrated entities and integration domains play
  a symmetric role (duality)
• Exhibits elegant connections with topology
  (homology / co-homology)

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3. ... What will we gain ? Global
parameterization methods
Ray et.al
Gu Yau
Tong et.al
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3. Gu and Yau's method
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3. Streamlines and beyond
Anisotropic Polygonal Remeshing Sig03
can we do a continuous version of this ?
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3. Periodic Global Parameterization
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3. Periodic Global Parameterization
Optimizes alignment with curvature tensor Demo
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3. Modified Tutte conditionSteiner Fischer
"Translational" (affine) Differential manifold
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3. Modified Tutte conditionDesbrun et. al
"Translational rot90" (complex) Differential
manifold
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4. Spectral Surface QuadrangulationDong et.al
Eigenfunction
Morse complex
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4. Spectral Surface QuadrangulationDong et.al
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4. Spectral methodsChladni Plates
sand
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4. Spectral methodsChladni Plates
Do re mi fa sol la si do
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4. Spectral methodsChladni Plates
Helmoltz wave eqn ? f l f
Discoveries concerning the theory of music
Chladni, 1787
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4. Spectral methodsLaplacian eigenfunctions
2D, square DCT basis (used by JPEG)
3D, sphere Spherical Harmonics
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4. Spectral MethodsWhat about 1D ?

• Equivalent to 0,2p with periodic conditions
  f(0) f(2p)
• ?f ?2f/?x2
• ?2 sin(wx)/?x2 -w2 sin(wx) (resp. cos)
• w needs to be an integer (periodic condition)

2p
0
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4. Spectral MethodsFourier basis for mesh
Reuter Shape DNA (eigenvalues as
signature) Zhang Shape Matching Demo
Manifold Harmonics
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Conclusion - Epilogue
PGP full automatic, but unelegant fix sing.
Quad design elegant, but partially manual
Spectral elegant and automatic, but does not
follow curvature tensor
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Conclusion - Speculations

• This leads to think that methods based on
  computing the eigenforms of some anisotropic
  differential operator may do the job
• Singularites of fractional index - spinors may be
  interesting to explore (but need to understand
  them)

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Followup

• Geometry Processing Based
• on Polygonal Meshes,
• Full-day course, Tuesday
• Organizers M. Botcsh, M. Pauly
• Speakers P. Alliez, L. Kobbelt, B. Lévy

You are here !!
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Resources

• Source code papers
• on http//alice.loria.fr
• Graphite
• OpenNL

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Calls for papers

• Eurographics 2008
• Abstracts Sept 21, papers Sept 26
• SPM / SPMI 2008
• Abstracts Nov 27, papers Dec 4
• SGP 2008
• Abstracts April 20, papers April 27
• Special issue Computing - eigenfunctions
• Abstracts Nov 1st, Papers Nov, 15

Paper copies of CfP available, ask us !
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Course Evaluations
4 Random Individuals will win an ATI Radeontm
HD2900XT
http//www.siggraph.org/course_evaluation