Manifold Harmonics: Laplacian Eigenfunctions for Computer Graphics

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Manifold Harmonics Laplacian
Eigenfunctionsfor Computer Graphics
Bruno Lévy INRIA, project ALICE
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Overview

• 1. Motivations Inspiration
• 2. The Discrete Setting
• 3. The Continuous Setting
• 4. Applications - Demos

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1. Motivations
Generate a "coordinate system"
LSCM in Maya
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1. MotivationsParameterization and Gridding
Constrained Parameterization Siggraph 1998 and
2001
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1. MotivationsArbitrary Topology Need for new
methods
Create a  geographic coordinate system 
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1. MotivationsGlobal Parameterizations
Gu Yau Global Conformal Param.
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1. Motivations - InspirationGlobal
Parameterizations
Dong et.al 2006 - Laplacian eigenfunction
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1. Motivations - Inspiration

• Signatures Reuter et. al Shape DNA
• Segmentation Liu Zhang
• Shape Correspondence Liu Zhang
• Shape Compression Karni Gotsmann

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2. The Discrete SettingGraph Laplacian
ai,j wi,j gt 0 if (i,j) is an edge ai,i -S
ai,j
(1,1 1) is an eigenvector assoc. with 0
The second eigenvector is interresting Fiedler
73, 75
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2. The Discrete SettingFiedler Vector
FEM matrix, Non-zero entries
Reorder with Fiedler vector
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2. The Discrete SettingFiedler Vector
Streaming meshes Isenburg Lindstrom
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2. The Discrete Setting Fiedler Vector
(geometrical interp.)
F(u) S wij (ui - uj)2
S ui 0
subject to
½ S ui2 1
L(u) ½ ut A u - l1 ut 1 - l2 ½ (utu - 1)
u eigenvector of A l1 0 l2 eigenvalue
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2. The Discrete Setting what about the other
eigenvectors ?

• ACE
• Local Linear Embedding
• Multi-Dimentional Scaling
• Manifold Learning
• Dimension Reduction
• Laplacian Eigenmaps
• Diffusion Wavelets
• ...

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3. The Continuous SettingChladni Plates
sand
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3. The Continuous SettingChladni Plates
Do re mi fa sol la si do
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3. The Continuous SettingChladni Plates
Helmoltz wave eqn ?2 f l f
Discoveries concerning the theory of music
Chladni, 1787
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3. The Continuous SettingFunctional Analysis
u v S ui vi
ltf,ggt ? f(t)g(t)dt
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3. The Continuous SettingFunctional Analysis
Operator L
Lf
f
eigenfunction
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3. The Continuous SettingFEM formulation

• Operator equation Lf lf
• L ? ?2./?x2 ?2./?y2
• Function basis (fi) f S ai fi
• Inner Product ltf,ggt ? f(x) g(x) dx
• ?i, ltLf, figt lltf, figt

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3. The Continuous SettingFEM Formulation

• Test function space (fi) f S ai fi
  (P1,P2,P3)
• ?i, ltDf, figt lltf, figt
• ltDf,ggt -lt?f,?ggt ( boundary term)
• Ax lBx
• aij -lt?fi,?fjgt bij ltfi,fjgt
• Note Lumped mass B diag( Sj bij )
• A B-1 A "discrete laplacian" used by
• geometry processing people.

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3. The Continuous SettingThe FEM Laplacian
G. Allaire Polytechnique course notes.
i
b
a
j
aij 2 (cotan a cotan b) / (Ai Aj)
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3. The Continuous SettingWhy this is important ?
FEM mesh Laplacian
Combinatorial mesh Laplacian
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3. The Continuous SettingNumerical Chladni Plate
Discrete Cosine Transform - JPEG
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3. The Continuous SettingSpherical Chladni
Spherical Harmonics
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3. The Continuous SettingMore complex objects
2D square Discrete Cosine
Transform Sphere Spherical
Harmonics Arbitrary shapes
An algorithm that "understands" geometry SMI 06
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3. The Continuous SettingWhy does it turn around
protrusions ?
Hindsight from the 1D circular case

• 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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3. The Continuous SettingMore geometric
properties
Nodal sets are sets of curves intersecting at
constant angles
The N-th eigenfunction has at most N eigendomains
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3. The Continuous SettingMore geometric
properties
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3. The Continuous SettingNumerical Method
1 million vertices, 1000 eigenfunctions ARPACK
TAUCS shift-invert
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3. The Continuous Setting Boundary Conditions
Neumann
Dirichlet
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3. The Continuous SettingHigher order function
basis
P1 function basis
P3 function basis
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4. Fourier Transformfor Meshes
Demo Interactive Convolution Filtering
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4. Finding the "Natural Parameters"
Demo "Manifold Harmonic Faces" Find the right
"tuning knobs" for a shape
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Conclusions

• An exciting research avenue
• Fourier Transform for Arbitrary Shapes
• Parameterize the function space on the shape
  rather than the shape
• Fast MHT (Fourier Transform) ?
• Diffusion wavelets Maggioni,
• Manifold Quasi-Harmonics

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Aknowledgements

• AIM_at_Shape, Microsoft, INRIA GEOREP
• Naoki Saito
• Matthias Zwicker (MIT), Ramsey Dyer (SFU)
• Michela Spanuolo (IMATI), Rhaleb Zayer (MPII)
• ALICE Researchers and Students

Bruno Vallet
Wan-Chiu Li
Nicolas Ray
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Anouncements ICIAM
minisymposium Geo-Topo methods for 3D shape
classification and matching Friday, 1115 -
1315 Room CHN-G-42 SIGGRAPH courses Mesh
parameterization Geometric Modeling Special
issue Computing computing-SI_at_eg.imati.cnr.it
ACM SPM (Symposium on Physical Modeling) IEEE
SMI (Shape Modeling International) June, 4-8,
Stony Brooks, USA http//www.cs.sunysb.edu/spm08
Abstracts Nov, 27 Full papers Dec,
4 Notification Jan, 31 links http//www.loria.f
r/levy