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Spherical Parameterization and Remeshing. Emil Praun, University of Utah Hugues Hoppe, Microsoft Research. Motivation: Geometry Images. [Gu et al. ’02]. completely regular sampling. 3D geometry. geometry image 257 x 257; 12 bits/channel. Motivation: Geometry Images.

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spherical parameterization and remeshing

Spherical Parameterization and Remeshing

Emil Praun, University of Utah

Hugues Hoppe, Microsoft Research

motivation geometry images
Motivation: Geometry Images

[Gu et al. ’02]

completely regular sampling

3D geometry

geometry image257 x 257; 12 bits/channel

motivation geometry images3
Motivation: Geometry Images
  • Geometry Images [Gu et al. ’02]
    • No connectivity to store
    • Render without memory gather operations
      • No vertex indices
      • No texture coordinates
    • Regularity allows use of image processing tools
spherical parametrization
Spherical Parametrization

Genus-0 models: no a priori cuts

geometry image257 x 257; 12 bits/channel

contribution
Contribution
  • Our method: genus-0  no constraining cuts
  • Less distortion in map; better compression
  • New applications:
    • morphing
    • GPU splines
    • DSP
outline
Outline
  • Spherical parametrization
  • Spherical remeshing
  • Results & applications
spherical parametrization8

sphere S

mesh M

Spherical Parametrization

[Kent et al. ’92]

[Haker et al. 2000]

[Alexa 2002]

[Grimm 2002]

[Sheffer et al. 2003]

[Gotsman et al. 2003]

  • Goals:
    • robustness
    • good sampling

 coarse-to-fine

 stretch metric

[Hoppe 1996]

[Hormann et al. 1999]

[Sander et al. 2001]

[Sander et al. 2002]

coarse to fine algorithm
Coarse-to-Fine Algorithm

Convert to progressive mesh

Parametrize coarse-to-fine

Maintain embedding & minimize stretch

coarse to fine algorithm10
Before Vsplit:

No degenerate/flipped 

 1-ring kernel 

Apply Vsplit:

No flips if V inside kernel

Coarse-to-Fine Algorithm

V

coarse to fine algorithm11
Before Vsplit:

No degenerate/flipped 

 1-ring kernel 

Apply Vsplit:

No flips if V inside kernel

Optimize stretch:

No degenerate  (they have  stretch)

Coarse-to-Fine Algorithm

V

traditional conformal metric
Traditional Conformal Metric
  • Preserve angles but “area compression”
  • Bad for sampling using regular grids
stretch metric
Stretch Metric

[Sander et al. 2001]

[Sander et al. 2002]

  • Penalizes undersampling
  • Better samples the surface
regularized stretch
Regularized Stretch
  • Stretch alone is unstable
  • Add small fraction of inverse stretch

without

with

outline15
Outline
  • Spherical parametrization
  • Spherical remeshing
  • Results & applications
domains and their sphere maps
Domains And Their Sphere Maps

tetrahedron

octahedron

cube

outline22
Outline
  • Spherical parametrization
  • Spherical remeshing
  • Results & applications
results26
Results

Model courtesy of Stanford University

David

timing results
Timing Results

Pentium IV, 3GHz, initial code

timing results28
Timing Results

Pentium IV, 3GHz, optimized code

rendering
Rendering

interpretdomain

rendertessellation

level of detail control
Level-of-Detail Control

n=1

n=2

n=4

n=8

n=16

n=32

n=64

morphing
Morphing
  • Align meshes & interpolate geometry images
geometry compression
Geometry Compression
  • Image wavelets
    • Boundary extension rules
      • spherical topology
      • Infinite C1 lattice*
    • Globally smooth parametrization*

*(except edge midpoints)

smooth geometry images
Smooth Geometry Images

[Losasso et al. 2003]

GPU

3.17 ms

33x33 geometry image

C1 surface

ordinary uniform bicubic B-spline

summary
Summary

original

sphericalparametrization

geometryimage

remesh

conclusions
Conclusions
  • Spherical parametrization
    • Guaranteed one-to-one
  • New construction for geometry images
    • Specialized to genus-0
    • No a priori cuts  better performance
    • New boundary extension rules
      • Effective compression, DSP, GPU splines, …
future work
Future Work
  • Explore DSP on unfolded octahedron
    • 4 singular points at image edge midpoints
  • Fine-to-coarse integrated metric tensors
    • Faster parametrization; signal-specialized map
  • Direct DSM optimization
  • Consistent inter-model parametrization