3 D Surface Parameterization

1. 3D Surface Parameterization Olga Sorkine, May 2005

2. Part OneParameterization and Partition Some slides borrowed from Pierre Alliez and Craig Gotsman

3. What is a parameterization? • S  R3 - given surface • D  R2 - parameter domain • s : D  S 1-1 and onto

4. Example – flattening the earth

5. Isoparametric curves on the surface • One parameter fixed, one varies: • Family 1 (varying u): Lv0 (u) = s(u, v0) • Family 2 (varying v): Mu0 (v) = s(v0, v)

6. Analytic example: Parameters: u = x, v = y D = [–1,1][–1,1]. z = z(x,y) = –(x2+y2) s(x,y) = (x, y, z(x,y))

7. h 1   -1 Another example: Parameters: , h D = [0,][–1,1] x(, h) = cos() y(, h) = h z(, h) = sin()

8. Triangular Mesh • Standard discrete 3D surface representation in Computer Graphics – piecewise linear • Mesh Geometry: list of vertices (3D points of the surface) • Mesh Connectivity or Topology: description of the faces

9. Triangular Mesh

10. Triangular Mesh

11. Mesh Representation Geometry: v1 – (x1, y1, z1) v2 – (x2, y2, z2) v3 – (x3, y3, z3) . . . vn – (xn, yn, zn) v3 v2 vn v1 Topology: Triangle list {v1, v2, v3} . . . {vk, vl, vm}

12. Mesh Parameterization • Uniquely defined by mapping mesh vertices to the parameter domain: U : {v1, …, vn} D  R2 U(vi) = (ui, vi) • No two edges cross in the plane (in D) Mesh parameterization  mesh embedding

13. Mesh parameterization Parameterizations EmbeddingU Parameter domain Mesh surface D  R2 S  R3 s = U -1

14. Mesh parameterization

15. Mesh parameterization s and U are piecewise-linear Linear inside each mesh triangle s U In2D In3D A mapping between two triangles is a uniqueaffine mapping

16. Barycentric coordinates C P A B

17. Mapping triangle to triangle s p3 q3 p1 q1 q2 p2

18. Only topological disks can be embedded • Other topologies must be “cut” or partitioned

19. Non-simple domains

20. Cutting

21. Applications of parameterization • Texture mapping • Surface resampling (remeshing) • Mesh compression • Multiresolution analysis Using parameterization, we can operate on the 3D surface as if it were flat

22. Texture mapping

23. Texture mapping

24. Texture mapping

25. Remeshing

26. Remeshing

27. Remeshing parameterization resampling

28. Remeshing

29. Remeshing examples

30. More remeshing examples

31. Bad parameterization…

32. Distortion measures • Angle preservation • Area preservation • Stretch • etc...

33. Bad parameterization

34. Better…

35. Distortion minimization Texture map Kent et al ‘92 Floater 97 Sander et al ‘01

36. Resampling on regular grid Resampling problems Cat mesh Distorting embedding

37. Dealing with distortion and non-disk topology Problems: 1) Parameterization of complex surfaces introduces distortion. 2) Only topological disk can be embedded. Solution: partition and/or cut the mesh into several patches, parameterize each patch independently.

38. Partition

39. Introducing seams (cuts)

40. Introducing seams (cuts)

41. Introducing seams (cuts)

42. Introducing seams (cuts)

43. Partition – problems • Discontinuity of parameterization • Visible artifacts in texture mapping • Require special treatment • Vertices along seams have several (u,v) coordinates • Problems in mip-mapping Make seams short and hide them

44. Piecewise continuous parameterization

45. Summary • “Good” parameterization = non-distorting • Angles and area preservation • Continuous param. of complex surfaces cannot avoid distortion. • “Good” partition/cut: • Large patches, minimize seam length • Align seams with features (=hide them)

46. End of Part One