Spherical Parameterization and Remeshing

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