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3 D Surface Parameterization

3 D Surface Parameterization. Olga Sorkine, May 2005. Part One Parameterization and Partition. Some slides borrowed from Pierre Alliez and Craig Gotsman. What is a parameterization?. S  R 3 - given surface D  R 2 - parameter domain s : D  S 1-1 and onto.

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3 D Surface Parameterization

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  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

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