1. Wavelets on Surfaces In partial fulfillment of the “Area Exam” doctoral requirements By Samson Timoner May 8, 2002 (picture from “Wavelets on Irregular Point Sets”)

2. Papers • Wavelets on Irregular Point Setsby Daubechies Guskov, Schroder and Sweldens(Trans. R. Soc. 1999) • “Spherical Wavelets: Efficiently Representation Functions on the Sphere” by Schroder and Sweldens • “The Lifting Scheme: Construction of second generation wavelets” by Sweldens • Multiresolution Signal Processing For Meshesby Guskov, Sweldens and Schroder(Siggraph 1999) • Multiresolution Hierarchies On Unstructured Triangle Meshes by Kobbelt, Vorsatz, and Seidel (Compu. Geometry: Theory and Applications, 1999)

3. Outline • Wavelets • The Lifting Scheme • Extending the Lifting Scheme • Application: Wavelets on Spheres • Wavelets on Triangulated Surfaces • Applications

4. Wavelets • Multi-resolution representation. • Basis functions (low pass filter). • Detail Coefficients (high pass filter). • We have bi-orthogonality between the detail coefficients and the basis-coefficients • Vanishing Moments

5. The Lifting Scheme Split

6. The Lifting Scheme Predict

7. The Lifting Scheme Predict

8. The Lifting Scheme Update 1/8{-1,2,6,2,-1}, ½{-1,2,-1}

9. The Lifting Scheme • Introduced “Prediction” • Translated and Scaled one filter. • We have bi-orthogonality between the detail coefficients and the basis-coefficients • 2 Vanishing Moments (mean and first) More Details

10. Irregularly Sampled Points Split Predict Update

11. Irregularly Sampled Points • Filters are no longer translations of each other. • Detail coefficients indicate different frequencies. • Perhaps it is wiser not to select every other point? • You can show bi-orthogonality(by vanishing moments).

12. Wavelets on Spheres • Sub-division on edges • Same steps • Split • Predict • Update

13. Topological Earth Data • Data is not smooth • All bases performed equally poorly. (picture from “Spherical Wavelets”) 15,000 coefficients 190,000 coefficients

14. Spherical Function: BRDF • Face Based methods are terrible (Haar-based) • Lifting doesn’t significantly help Butterfly. • Linear does better than Quadratic. 19, 73, 205 coefficients (pictures from “Spherical Wavelets”)

15. Up-Sampling Problems • Smooth interpolating polynomials • over-shooting • added undulations. • Linear interpolation isn’t smooth, but results are more intuitive.

16. Up-Sampling Problems • Similar problems can occur on surfaces. (picture from “Multiresolution Hierarchies On Unstructured Triangle Meshes”)

17. Wavelets on Spheres • Lessons: • Prediction is hard for arbitrary data sampling • Maybe lifting isn’t necessary for very smooth subdivision schemes? • Spheres are Special: • Clearly defined DC.(??zeroth order rep, smooth rep??) • Can easily make semi-regular mesh.

18. Outline • Wavelets: The Lifting Scheme • Wavelets on Triangulated Surfaces • Up-sampling problems • Applications

19. Triangulated Surfaces • “It is not clear how to design updates that make the [wavelet] transform numerically stable….” (Wavelets on Irregular Point Sets) • It is difficult to design filters which after iteration yield smooth surfaces. (Wim Sweldens in personal communication)

20. Lifting is hard • Prediction step is hard. • If you zero detail coefficients, you should get a “fair” surface. • Can’t use butterfly sub-division. • (picture from “Multiresolution Signal Processing For Meshes”)

21. Guskov et al. • Need Smoother as part of algorithm

22. Guskov et al. • Point Selection • Choose Smallest Edge • Remove one vertex in each level

23. Guskov et al. • Collapse the Edge

24. Guskov et al. • Prediction • Re-introduce the Edge. • Minimize Dihedral Angles • Detail Vector: Difference vector • (tangent plane coordinates)

25. Guskov et al. Quasi-Update • Smooth surrounding points (minimize dihedral angles)

26. Guskov et al. • Rough order of spatial frequencies. • Detail coefficients look meaningful. • Simple Smoothing: No “overshooting” errors. • No Guarantee of vanishing moments. • No Guarantee of bi-orthogonality. (picture from “Multiresolution Signal Processing For Meshes”)

27. Guskov et al. • Editing (picture from “Multiresolution Signal Processing For Meshes”)

28. Kobbelt et al. • Double Laplacian Smoother (thin plate energy bending minimization). • Solving PDE is slow! • Instead, solve hierarchically. (picture from “Multiresolution Hierarchies On Unstructured Triangle Meshes”)

29. Kobbelt et al. • Many vertices in each step (smallest edges first) • Prediction Step: location to minimize smoothing. • Detail: Perpendicular vector to local coordinate system. • Update: Smooth surrounding points

30. Kobbelt et al. • Rough order of spatial frequencies. • Fast: O(mn) with m levels, n verticies. • Many coefficients. • Bi-orthogonality? • Locality of filters? (picture from “Multiresolution Hierarchies On Unstructured Triangle Meshes”)

31. Are these wavelets? • Mathematically: No. • Bi-orthogonality • Too many coefficients.

32. Is this representation useful? • Patches do not wiggle; they remain in roughly the same position during down-sampling. • Smooth regions stay smooth. • Small detail coefficients. • Meaningful detail coefficients.

33. Outline • Wavelets: The Lifting Scheme • Wavelets on Triangulated Surfaces • Applications • Existing • Opportunities for new research

34. Editing • Replacing conventional surface editing. (NURBS) • (picture from “Multiresolution Signal Processing For Meshes” , • “Multiresolution Hierarchies On Unstructured Triangle Meshes”)

35. Feature Enhancement • “For show only.” • (picture from “Multiresolution Signal Processing for Meshes”)

36. Compression 549 Bytes(54e-4) 1225 Bytes(20e-4) 3037 Bytes(8e-4) 18111 Bytes(1.7e-4) Original • (picture from “Normal Mesh Compression”)

37. Remeshing • Go to low-resolution (to keep topology) and then sub-divide to restore original detail. • (picture from “Consistent Mesh Parameterizations”)

38. An Opportunity • Analysis of the wavelet coefficients

39. Statistics across Meshes • Use identical Triangulations across objects. • Look at statistics on detail coefficients rather than on points. • No global alignment problems. • No local alignment problems. • (I generated these images)

40. Feature Detection • Should be able to find signature hierarchical detail coefficients. • Hard with different triangulations. • (picture from “Multiresolution Signal Processing For Meshes” )

41. Acknowledgements • Professor White for suggesting the topic. • Wim Sweldens for responding to my e-mails. • Mike Halle and Steve Pieper for providing background information on the graphics community. Thank you all for coming today.

42. The Lifting Scheme Mathematics Low Pass Filter: 1/8(-1,2,6,2,-1) High Pass Filter: ½(-1,2,1) Back

43. Solving PDEs • Roughly, one can change the update and prediction step to have vanishing moments in the new orthogonality relationship.

44. Guskov et al. • Remove vertices in smoothest regions first. • Half-Edge Collapse to remove one vertex • Add vertex in, minimizing “second order difference”. • Smooth neighbors using same minimization • Detail coefficients are the movements between initial locations and final locations.

45. Kobbelt et al. • Select a fraction of the vertices. • Do half-edge collapses to remove the vertices. • Find a local parameterization around each vertex. • Add the vertex back in, minimizing the bending energy of the surface (Laplacian). • The detail vector is given by the coordinates of the point in the local coordinate system and a perpendicular height.

46. To Do List • Check Sphere coefficients • Sweldons Quote: change to published quote. • Edit Guskov et al • Compression Page: comments underneath.