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Spherical Parameterization and Remeshing

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

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  1. Spherical Parameterization and Remeshing Emil Praun, University of Utah Hugues Hoppe, Microsoft Research

  2. Motivation: Geometry Images [Gu et al. ’02] completely regular sampling 3D geometry geometry image257 x 257; 12 bits/channel

  3. 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

  4. Spherical Parametrization Genus-0 models: no a priori cuts geometry image257 x 257; 12 bits/channel

  5. Contribution • Our method: genus-0  no constraining cuts • Less distortion in map; better compression • New applications: • morphing • GPU splines • DSP

  6. Process

  7. Outline • Spherical parametrization • Spherical remeshing • Results & applications

  8. 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]

  9. Coarse-to-Fine Algorithm Convert to progressive mesh Parametrize coarse-to-fine Maintain embedding & minimize stretch

  10. Before Vsplit: No degenerate/flipped   1-ring kernel  Apply Vsplit: No flips if V inside kernel Coarse-to-Fine Algorithm V

  11. 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

  12. Traditional Conformal Metric • Preserve angles but “area compression” • Bad for sampling using regular grids

  13. Stretch Metric [Sander et al. 2001] [Sander et al. 2002] • Penalizes undersampling • Better samples the surface

  14. Regularized Stretch • Stretch alone is unstable • Add small fraction of inverse stretch without with

  15. Outline • Spherical parametrization • Spherical remeshing • Results & applications

  16. Domains And Their Sphere Maps tetrahedron octahedron cube

  17. Domain Unfoldings

  18. Boundary Constraints

  19. Spherical Image Topology

  20. Spherical Image Topology

  21. Spherical Image Topology

  22. Outline • Spherical parametrization • Spherical remeshing • Results & applications

  23. Example Results

  24. Results

  25. Results

  26. Results Model courtesy of Stanford University David

  27. Timing Results Pentium IV, 3GHz, initial code

  28. Timing Results Pentium IV, 3GHz, optimized code

  29. Rendering interpretdomain rendertessellation

  30. Level-of-Detail Control n=1 n=2 n=4 n=8 n=16 n=32 n=64

  31. Morphing • Align meshes & interpolate geometry images

  32. Geometry Compression • Image wavelets • Boundary extension rules • spherical topology • Infinite C1 lattice* • Globally smooth parametrization* *(except edge midpoints)

  33. 1.5 KB 3 KB 12 KB Compression Results

  34. Compression Results

  35. Smooth Geometry Images [Losasso et al. 2003] GPU 3.17 ms 33x33 geometry image C1 surface ordinary uniform bicubic B-spline

  36. Summary original sphericalparametrization geometryimage remesh

  37. 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, …

  38. 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

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