Manifold Parameterization

1. Manifold Parameterization Lei Zhang, Ligang Liu, Zhongping Ji, Guojin Wang Department of Mathematics Zhejiang University Accepted as regular paper by CGI2006

2. Overview • Parameterization • Least-squares Mesh • Manifold Parameterization • “Similar destination, different way” • “Similar way, different destination”

3. Reference • O. Sorkine and D. Cohen-Or. Least-squares meshes. In Proceedings of Shape Modeling International, 2004. • Lei Zhang, Ligang Liu, Zhongping Ji and Guojin Wang. Manifold Parameterization. Accepted as regular paper by ComputerGraphics International, 2006. • V. Kraevoy and A. Sheffer. Cross-Parameterization and Compatible Remeshing of 3D Models. SIGGRAPH, 2004.

4. Reference • M. Paone and Andrew Yuen. Mesh Fitting to Points. Projects Presentations (PPT), Simon Fraser University, Canada. • K. D. Cheng, W. P. Wang, H. Qin, K. K. Wong, H. P. Yang and Y. Liu. Fitting Subdivision Surfaces to Unorganized Point Data Using SDM. Proceedings of the 12th Pacific Conference on Computer Graphics and Applications, 2004.

5. Overview • Parameterization • Least-squares Mesh • Manifold Parameterization • “Similar destination, different way” • “Similar way, different destination”

6. Parameterization • Concept Parameterization is a one-to-one mapping from a triangular mesh surface onto a suitable domain. Domain Mesh

7. Planar Parameterization • Select a plane as the parameterization domain for an open mesh

8. Spherical Parameterization • Select a sphere as the parameterization domain for 0-genus mesh E. Praun and H. Hoppe, SIGGRAPH 04

9. Manifold Parameterization • Select a surface as parameterization domain for another surface Domain Mesh

10. Manifold Parameterization

11. Overview • Parameterization • Least-squares Mesh • Manifold Parameterization • “Similar destination, different way” • “Similar way, different destination”

12. Least-squares Meshes O. Sorkine, D. Cohen-Or Tel Aviv University Proceeding of Shape Modeling International 2004

13. Introduction • Mesh Mesh surface = Connectivity ? + Geometry ?

14. Introduction • Least-squares mesh Using a set of control points, approximate the original mesh surface by its connectivity graph.

15. Introduction 19851 vertices 200 control points 1000 control points 3000 control points

16. Least-squares meshes • Vertex conditions -Smooth condition L(vi)=0, vi all vertices -Geometry condition vj=cj, cj constraint • L-Laplacian operator Vi Vj

17. Least-squares meshes • Laplacian Equation • Smooth condition • Geometry condition vj=cj

18. Least-squares meshes • Example

19. Least-squares meshes • Equation Solution The system is solved in least-squares sense. A is sparse, and equation can be solved by TAUCS library quite fast.

20. Weighted Least-squares meshes • Higher weights for control points constraints

21. Weighted Least-squares meshes

22. Overview • Parameterization • Least-squares Mesh • Manifold Parameterization • “Similar destination, different way” • “Similar way, different destination”

23. Overview • Parameterization • Least-squares Mesh • Manifold Parameterization • “Similar destination, different way” • “Similar way, different destination”

24. Cross-Parameterization and Compatible Remeshing of 3D Models V. Kraevoy and A. Sheffer SIGGRAPH 2004

25. Introduction • Given two mesh M1 and M2, obtain correspondence via base meshes. f1 F f2-1 M2 M1 f1 f2 F B1 B2

26. Main Steps • Construct topologically identical path layouts • No interior intersection • Cyclical order • ……

27. Main Steps • Get topologically identical base mesh

28. Main Steps • Map patch layout to base mesh Mean value parameterization f1 f2

29. Main Steps • Construct mapping between base mesh F Barycentric coordinate

30. Main Steps • Result parameterization f1 F f2-1 M2 M1 f1 f2 F B1 B2

31. Examples

32. Conclusion • Indirect • Boring path layout searching • Time-consuming • ……

33. Overview • Parameterization • Least-squares Mesh • Manifold Parameterization • “Similar destination, different way” • “Similar way, different destination”

34. Mesh Fitting to Points M. Paone and A. Yuen Supervisor: Richard (Hao) Zhang Simon Fraser University, Canada Project Report

35. Fitting Subdivision Surfaces to Unorganized Point Data Using SDM K. D. Cheng, W.P. Wang, H. Qin, K. K. Wong, H. P. Yang and Y. Liu PG ’04

36. Introduction • Reconstruction of smooth surface from point clouds • Tool: Loop subdivision surface • Measure: SD (Squared Distance) H. Pottmann and M. Hofer. Geometry of the Squared Distance Function to Curves and Surfaces. Visualization and Mathematics III, Springer, 2003.

37. V2 V0 V1 V3 Loop Subdivision Edge-Vertex Vertex-Vertex

38. Squared Distance

39. Main Steps Normalization Target data points are scaled to .

40. Main Steps Normalization Pre-computation Compute distance field and curvatures at all data points.

41. Main Steps Normalization Pre-computation Initial mesh Use Marching Cubes to obtain an initial control mesh.

42. Main Steps Normalization Pre-computation Initial mesh Sampling Get sample points on limit surface. J. Stam. Evaluation of Loop Subdivision Surfaces. SIGGRAPH99, course.

43. Main Steps Normalization Pre-computation Initial mesh Optimization Sampling SDM error function:

44. Main Steps Normalization Pre-computation Initial mesh Error evaluation Optimization Sampling Maximum approximation error: Average error:

45. Main Steps Normalization Pre-computation Initial mesh Error evaluation Optimization Sampling Insert new control points to regions of large errors.

46. Examples

47. Thank You