1. Recent Progress in Mesh Parameterization Speaker : ZhangLei

2. Decade Retrospect 1997 1998 1999 20002001 2002 20032004200520062007 75 papers

3. Research Blocks Planar Parameterization MIPS, LSCM, Mean-value, ABF++,… Manifold Parameterization Spherical parameterization Simplex parameterization Inter-surface parameterization Volumetric Parameterization To be expected…

4. Alla Sheffer University of British Columbia ALICE, GEOMETRICA @INRIA Bruno LevyPierre Alliez M. S. Floater University of Oslo Craig Gotsman Israel Institute of Technology Hugues Hoppe Microsoft Research David Xianfeng Gu Stony Brook@State university of NewYOrk Kai Hormann Clausthal University of Technology

5. Curvilinear Spherical Prameterization Zayer, R. MPI Rossl, C. INRIA Seidel, H.P. MPI Shape Modeling and Applications, 2006

6. Curvilinear Coordinates System

7. Initial Parameterization pole date line pole

9. Secondary Parameterization Angle or Area Distortion Control

11. Local Domain Distortion Reduction Tangential Laplacian Smoothing

12. Results

13. Conclusion Pros Easy-to-implement Robust Cons Moderate distortion Poles and date lines selection

14. Linear Angle Based Parameterization Zayer, R. MPI Levy, B. INRIA-Alice Seidel, H. P. MPI Eurographics Symposium on Geometry Processing, 2007

15. ABF&ABF++ Sheffer, A., de Sturler, E. Parameterization of Faceted Surfaces for Meshing Using Angle Based Flattening. Engineering with Comp uters, 2001. Sheffer, A., Levy, B., Mogilnitsky, M., Bogomyakov, A. ABF++: Fast and Robust Angle Based Flattening. ToG, 2005. Coordinate space Angle space Coordinate space

16. ABF Planar Angle Constraints Vertex consistency Triangle consistency Wheel consistency

17. ABF Lagrange Multiplier Optimization Non-linear

18. Linearization Denote Logarithmic & Taylor expansion

19. linear

20. Linear ABF

21. Results

22. Conclusion ABF++ Pros & Linear computation

23. Discrete Conformal Mappings via Circle Patterns Kharevych, L. Caltech Springborn, B. TU Berlin Schroder, P. Caltech ACM Transactions on Graphics, 2006

24. Circle Packing combinatorics geometry William Thurston

25. THEOREM (The Dirichlet Problem) Let K be a complex trangulating a closed topological disc, let A be an angle sum target function of K, and assume that is a function defined on the boundary vertices of K. The n there exists a unique Euclidean packing label R for K with the property that for each boundary vertex. CirclePack http://www.math.utk.edu/~kens/

26. Circle Pattern

27. Circle Pattern Problem To reconstruct a circle pattern from an a bstract triangulation and the intersection angles.

28. Circle Pattern Delaunay Triangulation for interior edges for boundary edges Edge weight

29. Circle Pattern Delaunay Triangulation

30. Circle Pattern Problem Local Geometry of an Edge For a flat triangle

31. Variational Circle Patterns

32. Circle Pattern Problem vs Parameterization To reconstruct a circle pattern from an a bstract triangulation and the intersection angles. Discrete conformal parameterization of triangular mesh ?

33. Parameterization Algorithm 1. Setting the angles for each edge; 2. Minimizing the energy; 3. Generating the layout;

34. Parameterization Algorithm 1. Setting the angles for each edge; 2. Minimizing the energy; 3. Generating the layout;

35. Parameterization Algorithm 1. Setting the angles for each edge; 2. Minimizing the energy; 3. Generating the layout;

36. Results

37. Conclusion Pros Circle version of ABF … Cons Nonlinear

38. Periodic Global Parameterization Ray, N., Li, W. C., Levy, B. INRIA-Alice Sheffer, A. University of British Columbia Alliez, P. INRIA-Geometrica ACM Transactions on Graphics, 2006

39. Global Parameterization Given two charts C and C’, if their intersection is a topological disk, then the image of the intersection in param eterization space by and are linked by a geometric transition function : Translation: affine manifold General: complex manifold

40. Periodic Parameterization Rotation Translation

41. Problem Input: Output:

43. Formulation Objective

44. Transition function

45. Application Quad-Remeshing

46. Most Shape-preserving Mesh Parameteri- zation by Rigid Alignment

47. Complex Manifold Rigid transformation Translation Rotation

48. Local Shape-preserving Prameteriz ation 1-ring Patch: Geodesic Polar Map is an boundary vertex and otherwise

49. Global Shape-preserving Parameteriz ation Rigid Alignment

50. Least-squares Sense To minimize

51. Non-linear Configuration Solving by iterative linear system

52. Initial Parameterization For disc-like mesh with one boundary Floater, M. S. Parameterization and smoothing approximation of s urface triangulations. CAGD, 1997.

53. For mesh with multi-boundary Chen, Z. G., Liu, L. G., Zhang, Z. Y., Wang, G. J. Surface Paramete rization via aligning optimal local flattening. SPM, 2007.

54. Results General Curved Mesh

55. Developable Mesh

57. Thanks for your attention!

58. Planar Parameterization Disk-topology Mesh One-boundary Multi-boundary With constraints

59. Manifold Parameterization Spherical parameterization Simplex parameterization Inter-surface parameterization

60. Volumetric Parameterization