1. 3D Surface Parameterization Olga Sorkine, May 2005

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

3. What is a parameterization? S  R 3 - given surface S  R 3 - given surface D  R 2 - parameter domain D  R 2 - parameter domain s : D  S 1-1 and onto s : D  S 1-1 and onto

4. Example – flattening the earth

5. Isoparametric curves on the surface One parameter fixed, one varies: One parameter fixed, one varies: Family 1 (varying u): L v0 (u) = s(u, v 0 ) Family 1 (varying u): L v0 (u) = s(u, v 0 ) Family 2 (varying v): M u0 (v) = s(v 0, v) Family 2 (varying v): M u0 (v) = s(v 0, v)

6. Analytic example: Parameters: u = x, v = y D = [ – 1,1]  [ – 1,1]. z = z(x,y) = –(x 2 +y 2 ) s(x,y) = (x, y, z(x,y))

7. 1   h 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 linearStandard discrete 3D surface representation in Computer Graphics – piecewise linear Mesh Geometry: list of vertices (3D points of the surface) Mesh Geometry: list of vertices (3D points of the surface) Mesh Connectivity or Topology: description of the faces Mesh Connectivity or Topology: description of the faces

9. Triangular Mesh

11. Mesh Representation Geometry: v 1 – (x 1, y 1, z 1 ) v 2 – (x 2, y 2, z 2 ) v 3 – (x 3, y 3, z 3 )... v n – (x n, y n, z n ) Topology: Triangle list {v 1, v 2, v 3 }... {v k, v l, v m } v1v1v1v1 v2v2v2v2 v3v3v3v3 vnvnvnvn

12. Mesh Parameterization Uniquely defined by mapping mesh vertices to the parameter domain: Uniquely defined by mapping mesh vertices to the parameter domain: U : {v 1, …, v n }  D  R 2 U(v i ) = (u i, v i ) No two edges cross in the plane (in D ) No two edges cross in the plane (in D ) Mesh parameterization  mesh embedding

13. Mesh parameterization Parameter domain D  R 2 Mesh surface S  R 3 Embedding U Parameterization s s = U -1

14. Mesh parameterization

15. s and U are piecewise-linear Linear inside each mesh triangle In 2DIn 3D U s A mapping between two triangles is a unique affine mapping

16. A B C P Barycentric coordinates

17. Mapping triangle to triangle s p1p1 p2p2 p3p3 q1q1 q2q2 q3q3

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

19. Non-simple domains

20. Cutting

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

22. 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 preservationAngle preservation Area preservationArea preservation StretchStretch etc...etc...

33. Bad parameterization

34. Better…

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

36. Resampling problems Cat meshDistorting embedding Resampling on regular grid

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)

43. Partition – problems Discontinuity of parameterization Discontinuity of parameterization Visible artifacts in texture mapping Visible artifacts in texture mapping Require special treatment 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 “Good” parameterization = non-distorting –Angles and area preservation –Continuous param. of complex surfaces cannot avoid distortion. “Good” partition/cut: “Good” partition/cut: –Large patches, minimize seam length –Align seams with features (=hide them)

46. End of Part One