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

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

3. Geometry Images [Gu et al. ’02] No connectivity to store No connectivity to store Render without memory gather operations Render without memory gather operations –No vertex indices –No texture coordinates Regularity allows use of image processing tools Regularity allows use of image processing tools Motivation: Geometry Images

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

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

6. Process

7. Outline 1.Spherical parametrization 2.Spherical remeshing 3.Results & applications

8. Spherical Parametrization Goals: robustness robustness good sampling good sampling sphere S mesh M [Sander et al. 2001] [Hormann et al. 1999] [Sander et al. 2002] [Hoppe 1996]  coarse-to-fine  stretch metric  coarse-to-fine  stretch metric [Kent et al. ’92] [Haker et al. 2000] [Alexa 2002] [Grimm 2002] [Sheffer et al. 2003] [Gotsman et al. 2003]

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

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

11. Before Vsplit: No degenerate/flipped  No degenerate/flipped   1-ring kernel  Apply Vsplit: No flips if V inside kernel Optimize stretch: No degenerate  (they have  stretch) V Coarse-to-Fine Algorithm

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 withoutwith

15. Outline 1.Spherical parametrization 2.Spherical remeshing 3.Results & applications

16. Domains And Their Sphere Maps tetrahedron octahedron cube

17. Domain Unfoldings

18. Boundary Constraints

19. Spherical Image Topology

22. Outline 1.Spherical parametrization 2.Spherical remeshing 3.Results & applications

23. Example Results

24. Results

26. David Model courtesy of Stanford University

27. Timing Results Model # faces Time Cow23,216 7 min. 7 min. David60,000 8 min. Bunny69,630 10 min. Horse96,948 15 min. Gargoyle200,000 23 min. Tyrannosaurus200,000 25 min. Pentium IV, 3GHz, initial code

28. Timing Results Model # faces Time Cow23,216 65 sec. David60,000 80 sec. Bunny69,630 1.5 min. Horse96,948 2.5 min. Gargoyle200,000 4 min. Tyrannosaurus200,000 Pentium IV, 3GHz, optimized code

29. Rendering interpret domain render tessellation

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 Boundary extension rules –spherical topology –Infinite C 1 lattice* Globally smooth parametrization* Globally smooth parametrization* *(except edge midpoints)

33. Compression Results 12 KB3 KB1.5 KB

34. Compression Results

35. Smooth Geometry Images 33x33 geometry image C 1 surface GPU 3.17 ms [Losasso et al. 2003] ordinary uniform bicubic B-spline

36. Summary original spherical parametrization geometry image remesh

37. Conclusions Spherical parametrization Guaranteed one-to-one Guaranteed one-to-one New construction for geometry images Specialized to genus-0 Specialized to genus-0 No a priori cuts  better performance No a priori cuts  better performance New boundary extension rules New boundary extension rules –Effective compression, DSP, GPU splines, …

38. Future Work Explore DSP on unfolded octahedron 4 singular points at image edge midpoints 4 singular points at image edge midpoints Fine-to-coarse integrated metric tensors Faster parametrization; signal-specialized map Faster parametrization; signal-specialized map Direct D  S  M optimization Consistent inter-model parametrization