Transcript Mesh Parameterization: Theory and Practice Global Methods Bruno Lévy - INRIA Overview 1. Introduction - Motivations - Difficulties 2.

Mesh Parameterization:
Theory and Practice
Global Methods
Bruno Lévy - INRIA
Overview
1. Introduction - Motivations - Difficulties
2. Base Complex
3. Exterior Calculus
4. Spectral methods
Epilogue / Conclusion
1. Motivations
Generate a "coordinate system"
Texture
atlas
1. Motivations
Arbitrary Topology : Need for new methods
Create a « geographic coordinate system »
1. Motivations
Remeshing and Geometry Images
1. Motivations
Reverse engineering
1. Difficulties
Singularities
1. Difficulties
Singularities
Archibald Higgins Adventures (used with author permission)
For french readers, Archibald is known as Anselme Lanturlu
http://www.savoir-sans-frontieres.org
1. Difficulties
Singularities
1. Difficulties
Singularities
1. Difficulties
Singularities
Poincarré Hopf
DEC version:
N-Symmetry
direction fields
[Ray et.al]
See also N-Rosy
[Zhang et.al]
1. Difficulties
Functions over manifolds
Differential Manifold
see, e.g. [Grimm et.al], [Gortler et.al], [Ray et.al]
2. Base Complex
Triangulated Base Complex
Multires Analysis, MAPS,
Inter-Chart Relaxation
2. Base Complex
Quadrangulated Base Complex
Polycubemaps
[Tarini et.al]
Semi-manual construction
2. Base Complex
Why quads are desirable ?
2. Base Complex
Frustration with charts
IP =
x
u
2
x x
u v
x x
u v
x
v
2
Lots of computations !
Handling charts is a hassle
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

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)
3. ... What will we gain ?
Global parameterization methods
[Ray et.al]
[Gu & Yau]
[Tong et.al]
3. [Gu and Yau's] method
3. Streamlines and beyond
Anisotropic
Polygonal
Remeshing
[Sig03]
can we do a
continuous
version of this ?
3. Periodic Global Parameterization
q
cos(q)
U=
sin(q)
3. Periodic Global Parameterization
Optimizes alignment with curvature tensor
Demo
3. Modified Tutte condition
[Steiner & Fischer]
"Translational" (affine)
Differential manifold
3. Modified Tutte condition
[Desbrun et. al]
"Translational + rot90"
(complex)
Differential manifold
4. Spectral Surface Quadrangulation
[Dong et.al]
Eigenfunction
Morse complex
4. Spectral Surface Quadrangulation
[Dong et.al]
4. Spectral methods
Chladni Plates
sand
4. Spectral methods
Chladni Plates
Do re mi fa sol la si do …
4. Spectral methods
Chladni Plates
Helmoltz wave eqn: Δ f = l f
Discoveries concerning the theory of music Chladni, 1787
4. Spectral methods
Laplacian eigenfunctions
2D, square
DCT basis
(used by JPEG)
3D, sphere
Spherical
Harmonics
4. Spectral Methods
What about 1D ?




Equivalent to [0,2p] with periodic conditions f(0) = f(2p)
Δf = ∂2f/∂x2
∂2 sin(wx)/∂x2 = -w2 sin(wx)
(resp. cos)
w needs to be an integer (periodic condition)
0
2p
4. Spectral Methods
Fourier basis for mesh
[Reuter] Shape DNA (eigenvalues as signature)
[Zhang] Shape Matching
Demo: Manifold Harmonics
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
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)
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 !!
Resources

Source code & papers
on http://alice.loria.fr
– Graphite
– OpenNL
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 !
Course Evaluations
4 Random Individuals will win an ATI Radeontm HD2900XT
http://www.siggraph.org/course_evaluation