Recent Progress in Mesh Parameterization

1. Recent Progress in Mesh Parameterization Speaker : ZhangLei

2. Decade Retrospect 75 papers 2000 2001 2002 2003 2004 2005 2006 2007 1997 1998 1999

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

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 Levy, B. INRIA-Alice Seidel, H. P. MPI Zayer, R. 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 Computers, 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 Springborn, B. TU Berlin Kharevych, L. Caltech Schroder, P. Caltech ACM Transactions on Graphics, 2006

24. Circle Packing William Thurston combinatorics geometry

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. Then 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 abstract triangulation and the intersection angles.

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

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 abstract triangulation and the intersection angles. Discrete conformal parameterization of triangular mesh ?

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

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

35. Parameterization Algorithm Setting the angles for each edge; Minimizing the energy; 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 parameterization space by and are linked by a geometric transition function : Translation: affine manifold General: complex manifold

40. Periodic Parameterization Translation Rotation

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 Prameterization • 1-ring Patch: Geodesic Polar Map is an boundary vertex and otherwise

49. Global Shape-preserving Parameterization • Rigid Alignment

50. Least-squares Sense To minimize