Mesh Parameterizations

1. Mesh Parameterizations Lizheng Lu Lulz_zju@yahoo.com.cn Oct. 19, 2005

2. Overview • Introduction • Planar Methods • Non-Planar Methods • Mean Value Methods • Spherical Methods • Summary

3. Motivation(1) • Analysis on surfaces is usually performed in Eucli- dean plane, using appropriate (local) coordinates. • ⇒One has to assign to every surface point a parameter value in the plane • The result of the analysis often depends on the choice of the parameterization.

4. Example: B-Spline Interpolation

5. Motivation(2) Q: What is a good parameterization ? A: One that preserve all the (basic) geometry length, angles, area, ... ⇒ isometric parameterization but : possible only for developable surfaces e.g., there will always be distortion ! Try to keep the distortion as small as possible (change of length, area, angles,... )

6. Motivation(3): Applications Many operations, manipulations on/with surfaces require a parameterization as a preliminary step. e.g.: Texture mapping Surface fitting Hierarchical representations Mesh conversion Morphing & Deformation

7. Motivation: Applications of Parameterizations

8. Motivation: Applications of Parameterizations

9. Motivation: Applications of Parameterizations

10. Motivation: Applications of Parameterizations Morphing

11. Problem Description For a triangulated set of data points find a parameteration with minimal distortion

12. Classifications • Conformal mapping • No distortion in angles • Equiareal mapping • No distortion in areas • Isometric mappings • No distortion, but usually impossible

13. Desirable Properties • With minimal distortion • So how to measure and minimize it? • Guaranteeing one-to-one mapping • Avoid overlapping, degenerating, flipping • Most difficult and critical! • Robustness • Time and space efficiency • Process meshes with genus, if possible

14. Previous Methods:Classifications • Planar methods • Early works • Cube/Polycube methods • Spherical methods • Partition methods • … Goal: Minimizing distortion for diverse meshes

15. Overview • Introduction • Planar Methods • Non-Planar Methods • Mean Value Methods • Spherical Methods • Summary

16. Theory Foundation • Given: • A planar 3-connected graph • Boundary fixed to a convex shape in R2 • Result: • Interior vertices form a planar triangular • Each vertex is some convex combination of its neighbors

17. Main Challenge • Measure of distortion • Ratio of singular values (Hormann & Greiner 1998) • Conformal (Levy, 2002) • Dirichlet energy (Guskov, 2002) • Mean value (Floater, 2003,2005 & Tao Ju,2005) • Boundary fixing • Choose the shape, e.g.. Circle, square, etc. • Choose the distribution • Seamless merging • Partition and cutting

18. Main References M. S. Floater. Parameterization and smooth approximation of surface triangulations. CAGD , 1997, 14(3):231-250. Hormann, Greiner: MIPS: An efficient global parametrization method, in: Curve and Surface Design: Saint−Malo1999,153−162 M. S. Floater and M. Reimers. Meshless parameterization and surface reconstruction. CAGD , 2001, 18(2):77-92. M. S. Floater, Mean value coordinates, CAGD , 2003,20(1), 19-27. M. S. Floater, One-to-one piecewise linear mappings over triangulations, Math. Comp. 2003,72(242), 685-696. M. S. Floater and K. Hormann, Surface Parameterization: a Tutorial and Survey, in Advances in Multiresolution for Geometric Modelling, N. A. Dodgson, M. S. Floater, and M. A. Sabin (eds.), Springer-Verlag, Heidelberg, 2004, 157-186.

19. Fixing the boundary of the mesh onto an unit circle an unit square Linear Methods: Idea

20. Linear Methods: Idea For interior mesh points: ⇒ Forming a linear system.

21. Choices of the Weights • Uniform: • Chord length: • Centripetal: • Mean value:

22. Shortcomings • Severe distortion • Topology limiting • Can't process non genus-zero meshes • Introduce other artifacts • Such as cutting seams

23. Non-linear Methods • [Hormann et al. 1999] MIPS • [Piponi et al. 2000] Seamless texture mapping of subdivision surfaces by model pelting and texture blending. SIGGRAPH • [Sander et al. 2001] Texture mapping progressive meshed. SIGGRAPH • [Zigelman et al. 2001] Texture mapping using surface flattening via multi-dimensional scaling. TVCG, 8(2), 198-207

24. Overview • Introduction • Planar Methods • Non-Planar Methods • Mean Value Methods • Spherical Methods • Summary

25. How to Obtain Good Parameterization • Mesh independence? • Very difficult • Less distortion? • Maybe, defining better measure function • Possible method for minimizing distortion • Choosing possible mapping domains! • Sphere, Cube/polycub, Simplified domains • ...

26. Main References(1)Spherical Domain • Sheffer, A., Gotsman, C., Dyn, N. 2004. Robust Spherical Parameterization of Triangular Meshes. Computing, 72(1-2), 185–193. • Praun, E., Hoppe, H. Spherical Parametrization and Remeshing. SIGGRAPH2003. • Gotsman, C., Gu, X., Sheffer, A. Fundamentals of Spherical Parameterization for 3D Meshes. SIGGRAPH 2003. • Alexa, M. Recent advances in mesh morphing. 2002. Computer Graphics Forum, 21(2), 173-196. • Grimm, C. Simple manifolds for surface modeling and parametrization. Shape Modeling International 2002. • Haker, S., Angenent, S., et al. Conformal surface parameterization for texture mapping. 2000. TVCG, 6(2), 181-189. • Kobbelt, L.P., Vorsatz, J., Labisk, U., Seidel, J.-p.. A shrink-wrapping approach to remeshing polygonal surfaces. 1999. CGF. 18(3), 119-129. • Kent, J., Carlson, W., Parent, R. 1992. Shape transformation for polyhedral objects. SIGGRAPH 1992, 47-54.

27. Main References(2)Cube/Polycube Domain • Tarini, M., Hormann, K., Cigononi, P., Montani, C. PolyCube-Maps. SIGGRAPH 2004.

28. Main References(3) Simplified Domains • Schreiner, J., Asirvatham, A, Praun, E., Hoppe, H. Inter-Surface Mapping. SIGGRAPH 2004. • Khodakovsky, A., Litke, N., Schröder, P. Globally SmoothParameterizations with Low Distortion. SIGGRAPH 2003. • Gu, X., Gortler, J., Hoppe, H. Geometry images. SIGGRAPH 2002. • Sorkine, O., Cohen-or, D., et al. Bounded-distortion piecewise mesh parametrization. 2002. IEEE Visualization, 355-362. • Praun, E. Sweldens, W. Schröder, P. Consistent mesh parametrizations. SIGGRAPH 2001. • Guskov, I., Vidimce, K., Sweldens, W., Schröder, P. Normal meshes. SIGGRAPH 2000. • Lee, A., Dobkin, D., Sweldens, W., Schröder, P. Multiresolution mesh morphing. SIGGRAPH 1999. • Hoppe, H. Progressive meshes. SIGGRAPH 1996, 99-108.

29. Example Spherical Methods

30. ExamplePolycube Methods

31. ExampleSimplification/Cutting Methods

32. Overview • Introduction • Planar Methods • Non-Planar Methods • Mean Value Methods • Spherical Methods • Summary

33. Mean Value Coordinates for Closed Triangular Mesh Tao Ju, Scott Schaefer, Joe Warren Rice University SIGGRAPH2005

34. Mean Value MethodsReferences • M. S. Floater. Mean value coordinates. CAGD, 14(3):19–27, 2003. • M. S. Floater. Mean value coordinates in 3D. CAGD, 22(7):623–631, 2005. • Tao, Ju Scott Schaefer, Joe Warren. Mean Value Coordinates for Closed Triangular Meshes. SIGGRAPH 2005.

35. Barycentric Coordinates Give , find weights such that with barycentric coordinates

36. Boundary Value Interpolation • Given , compute such that • Given values at , construct a function • Good properties: • Interpolates values at vertices • Linear on boundary • Smooth on interior

37. Continuous Barycentric Coordinates Discrete Continuous

38. Mean Value Interpolation • Properties: • Interpolates boundary • Generates smooth function • Reproduces linear functions

39. Relation to Discrete Coordinates MV coordinates ⇒ Closed-form solution of continuous interpolant for piecewise linear shapes DiscreteContinuous

40. 3D Mean Value Coordinates • Project surface onto sphere centered at v • m = mean vector (integral of unit normal over spherical triangle) • Stokes’ Theorem:

41. Computing the Mean Vector • Given spherical triangle, compute mean vector (integral of unit normal) • Build wedge with face normals • Apply Stokes’Theorem,

42. Interpolant Computation • Compute mean vector: • Calculate weights • By • Sum over all triangles

43. Implementation Considerations • Special cases • On boundary (co-planar) • Numerical stability • Small spherical triangles • Large meshes • Pseudo-code provided in paper

44. ApplicationsBoundary Value Problems

45. ApplicationsSolid Textures

46. Real-time! Initial mesh Applications Surface Deformations

47. Summary • Integral formulation for closed surfaces • Closed-form solution for triangle meshes • Numerically stable evaluation • Applications • Boundary Value Interpolation • Volumetric textures • Surface Deformation

48. Overview • Introduction • Planar Methods • Non-Planar Methods • Mean Value Methods • Spherical Methods • Summary

49. Challenges on Spherical Domain • Robustness • Non-overlapping -->> Difficult and critical • 1-to-1 spherical map -->> Required • Less distortion • Diverse meshes -->> Highly deformed • Oversampling/downsamping -->>Irregular • So, how to miminize it? • Visually pleasing, regular, …

50. Previous Spherical Methods(1) • [Kent et al. 92]Shape Transformation for Polyhedral Objects. SIGGRAPH. • Projections Methods: • Convex and Star-Shaped Objects • Methods using model knowledge • Physically-Based Simulation • Simulating balloon inflation process • Hybrid methods • Unsolved problem, 1-to-1 map?…