Multiresolution Analysis of Arbitrary Meshes

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Multiresolution Analysis of Arbitrary Meshes

• Matthias Eck
• joint with
• Tony DeRose, Tom Duchamp, Hugues Hoppe,
• Michael Lounsbery and Werner Stuetzle

U. of Darmstadt , U. of Washington , Microsoft ,
Alias
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Overview

• 1. Motivation and applications
• 2. Our contribution
• 3. Results
• 4. Summary and future work

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Motivation

• problem complex shapes complex meshes

I have 70,000 faces !
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• Difficulties
• Storage
• Transmission
• Rendering
• Editing
• Multiresolution analysis

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• multiresolution representation of mesh M
• base shape M 0
• sum of local correction terms
• (wavelet terms)

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base shape M 0
mesh M
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Applications
1. Compression 2. Multiresolution editing 3.
Level-of-detail control 4. Progressive
transmission and rendering
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e lt 0.8
70,000 faces 11,000
faces
tight error bounds
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Applications
1. Compression 2. Multiresolution editing 3.
Level-of-detail control 4. Progressive
transmission and rendering
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Applications
1. Compression 2. Multiresolution editing 3.
Level-of-detail control 4. Progressive
transmission and rendering
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no visual discontinuties
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Applications
1. Compression 2. Multiresolution editing 3.
Level-of-detail control 4. Progressive
transmission and rendering
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base shape M 0
mesh M
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2. Our contribution
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2. Our contribution
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Previous work

• Lounsbery, DeRose, Warren 1993
• provides general framework for MRA
• extends wavelet analysis to surfaces of arbitrary
  topology
• Schroeder, Sweldens 1995
• similar work on sphere

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• However ...
• input surface must be parametrized over a simple
  domain mesh

• x

• r(x)

r
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The problem ...

• Meshes are typically given as collection of
  triangles, thus
• MRA algorithms cannot directly be applied

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Im not parametrized !
M
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... and our solution

• step 1 construct a simple domain mesh K

K
M
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... and our solution

• step 1 construct a simple domain mesh K
• step 2 construct a parametrization r of M over K

MRA !!!
r
K
M
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step1Construction of domain mesh

• Main idea
• partition M into triangular regions
• domain mesh K

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mesh M
partition
domain mesh K
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How to get partition ?

• Our requirements
• topological type of K topological type of M
• small number of triangular regions
• smooth and straight boundaries
• fully automatic procedure

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construct Voronoi-like diagram on M
construct Delaunay-like triangulation
mesh M
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step 2Construction of parametrization

• map each face of domain mesh to corresponding
  triangular region
• local maps agree on boundaries

parametrization r
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local map
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How to map locally?

• Requirements
• fixed boundary conditions
• small distortion
• Best choice harmonic maps
• well-known from differential geometry
• minimizing the metric distortion

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local map
planar triangle
triangular region
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4. Results
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4. Results
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34 min. , 70,000 faces
162 faces
2,000 faces , e lt 2.0
4,600 faces , e lt 1.2
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40 min. , 100,000 faces
229 faces
2,000 faces , e lt 2.0
4,700 faces , e lt 1.5
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Summary

• Given An arbitrary mesh M
• We construct a simple domain mesh and an exact
  parametrization for M
• Allows MRA to be applied
• tight error bounds
• Useful in other applications

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5. Future work
Other potential applications of parametrization

• texture mapping
• finite element analysis
• surface morphing
• B-spline fitting

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B - spline fitting
approximating surface
B - spline control mesh