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Spring 2015 CSCI 599: Digital Geometry Processing 8.1 Surface Parameterization Hao Li http://cs599.hao-li.com 1 Modeling 2 Modeling 3 Viewpaint The creation of a 3D assets surface, including that surfaces color, texture, opacity, and


  1. Spring 2015 CSCI 599: Digital Geometry Processing 8.1 Surface Parameterization Hao Li http://cs599.hao-li.com 1

  2. Modeling 2

  3. Modeling 3

  4. Viewpaint The creation of a 3D assets surface, including that surface’s color, texture, opacity, and reflectivity (or specularity). 4

  5. Viewpaint Rango: Creating creature scale textures in ZBrush... 5

  6. Viewpaint (Wrinkle Pass) 6

  7. Color Maps 7

  8. Wet Maps 8

  9. bump Maps 9

  10. Motivation Texture Mapping Levy et al.: Least squares conformal maps for automatic texture atlas generation , SIGGRAPH 2002. 10

  11. Motivation Normal Mapping 11

  12. Motivation 12

  13. Motivation 13

  14. Motivation 14

  15. Mesh Parameterization Find a 1-to-1 mapping between given surface mesh and 2D parameter domain 15

  16. Unfolding Earth A C B D C A B D 16

  17. Spherical Coordinates π φ φ θ −π θ 0 2 π � θ ⇤ ⌅ sin θ sin φ ⇥ cos θ sin φ ⇥� ⇧ ⌃ φ cos φ 17

  18. Desirable Properties Low distortion Bijective mapping 18

  19. Cartography orthographic stereographic Mercator Lambert preserves angles � preserves area � = conformal = equiareal Floater, Hormann: Surface Parameterization: A Tutorial and Survey , Advances in Multiresolution for Geometric Modeling, 2005 19

  20. More Maps 20

  21. Demo: Parameterization 21

  22. Recall: Differential Geometry Parametric surface representation x : Ω � IR 2 S � IR 3 ⇥ x u x v   x ( u, v ) p y ( u, v ) ( u, v ) ⇤⇥   z ( u, v ) Regular if � • Coordinate functions x,y,z are smooth • Tangents are linearly independent x u � x v ⇥ = 0 22

  23. Definitions A regular parameterization is � x : Ω → S • Conformal (angle preserving), if the angle of every pair of S intersecting curves on is the same as that of the Ω corresponding pre-images in . • Equiareal (area preserving) if every part of is mapped Ω S onto a part of with the same area • Isometric (length preserving), if the length of any arc on S Ω is the same as that of its pre-image in . 23

  24. Distortion Analysis x : IR 2 → IR 3 u = ( u, v ) x = ( x, y, z ) Jacobian transforms infinitesimal vectors   x u x v d x = J d u J = y u y v   z u z v � d x � 2 = (d u ) T J T J d u = (d u ) T I d u 24

  25. First Fundamental Form Characterizes the surface locally � x T x T ⇥ u x u u x v I = x T x T u x v v x v Allows to measure on the surface � d u T � ⇥ cos θ = 1 I d u 2 / ( ⇥ d u 1 ⇥ · ⇥ d u 2 ⇥ ) • Angles d s 2 = d u T I d u • Length d A = det( I ) d u d v • Area 25

  26. Isometric Maps x ( u, v ) A regular parameterization is isometric, iff its first fundamental form is the identity: � ⇥ 1 0 I ( u, v ) = 0 1 A surface has an isometric parameterization iff it has zero Gaussian curvature 26

  27. Cylinder 27

  28. Conformal Maps (A-Similar-AP) x ( u, v ) A regular parameterization is conformal, iff its first fundamental form is a scalar multiple of the identity: � ⇥ 1 0 I ( u, v ) = s ( u, v ) · 0 1 f 28

  29. Conformal Flow Crane et al. Spin Transformations of Discrete Surfaces, ACM Siggraph 2011 29

  30. Equiareal Maps x ( u, v ) A regular parameterization is equiareal, iff the determinant of its first fundamental form is 1: det( I ( u, v )) = 1 30

  31. Relationships An isometric parameterization is conformal and equiareal, and vice versa: isometric ⇔ conformal + equiareal Isometric is ideal, but rare. In practice, people try to compute: � • Conformal • Equiareal • Some balance between the two 31

  32. Harmonic Maps • A regular parameterization is harmonic, iff it x ( u, v ) satisfies ∆ x ( u, v ) = 0 • isometric ⇒ conformal ⇒ harmonic • Easier to compute than conformal, but does not preserve angles 32

  33. Harmonic Maps • A harmonic map minimizes the Dirichlet energy Z Z kr x k 2 = k x u k 2 + k x v k 2 d u d v Ω Ω • Variational calculus then tells us that ∆ x ( u, v ) = 0 • If is harmonic and maps the boundary of x : Ω → S ∂ Ω Ω ⊂ R 2 a convex region homeomorphically onto the boundary , then is one-to-one. ∂ S x 33

  34. Parameterization Goal • Piecewise linear mapping of a discrete 3D triangle mesh onto a planar 2D polygon • Slightly different situation: Given a 3D mesh, compute the inverse parameterization 34

  35. Floater’s Parameterization 35

  36. Floater’s Parameterization • For Quadrilateral Patch • Fix the parameters of the boundary vertices on a unit square • Derive the bijection for each of the interior vertices u v i by solving 36

  37. Floater’s Algorithm • Compute for each the i • Compute a local parameterization for that v ( i ) preserves the aspect ratio of the angle and length • Compute that satisfies � � • Solve the sparse equation for u ( v i ) , i = 1 . . . n 37

  38. Discrete Harmonic Maps • Map the boundary homeomorphically to some ∂ S (convex) polygon in the parameter plane ∂ Ω • Minimize the Dirichlet energy of by solving the u corresponding Euler-Lagrange PDE ∆ S u = 0 • Requires discretization of Laplace-Beltrami • Compare to surface fairing 38

  39. Discrete Harmonic Maps • System of linear equations � ∀ v i ∈ S : w ij ( u ( v j ) − u ( v i )) v j ∈ N 1 ( v i ) α ij w ij = cot α ij + cot β ij v i v j β ij • Properties of system matrix: • Symmetric + positive definite → unique solution • Sparse → efficient solvers 39

  40. Discrete Harmonic Maps • But… • Does the same theory hold for discrete harmonic maps as for harmonic maps? • In other words, is it possible for triangles to flip or become degenerate? 40

  41. Convex Combination Maps • If the linear equations are satisfied � w ij ( u ( v j ) − u ( v i )) � v j ∈ N 1 ( v i ) � and if the weights satisfy � � w ij > 0 w ij = 1 ∧ v j ∈ N 1 ( v i ) � then we get a convex combination mapping. 41

  42. Convex Combination Maps • Each is a convex combination of u ( v j ) u ( v i ) � u ( v i ) = w ij u ( v j ) v j ∈ N 1 ( v i ) • If is a convex combination map that u : S → Ω maps the boundary homeomorphically to the ∂ S Ω ⊂ R 2 boundary of a convex region , then ∂ Ω u is one-to-one. 42

  43. Convex Combination Maps • Uniform barycentric weights w ij = 1 / valence( v i ) • Cotangent weights ( if ) α ij + β ij < π > 0 w ij = cot( α ij ) + cot( β ij ) α ij δ ij v i γ ij • Mean value weights v j β ij w ij = tan( δ ij / 2) + tan( γ ij / 2) ⇥ p j � p i ⇥ (no negative weights, even for obtuse angles) 43

  44. Convex Combination Maps • Comparison original uniform mean cotan mesh weights value weights (shape preserving) 44

  45. Fixing the Boundary • Choose a simple convex shape • Triangle, square, circle • Distribute points on boundary • Use chord length parameterization Fixed boundary can create high distortion 45

  46. Open Boundary Mappings • Include boundary vertices in the optimization • Produces mappings with lower distortion 46

  47. Open Boundary Mappings 47

  48. Need disk-like topology • Introduce cuts on the mesh 48

  49. Naive Cut, Numerical Problems 49

  50. Smart Cut, Free Boundary 50

  51. Texture Atlas Generation • Split model into number of patches (atlas) • because higher genus models cannot be mapped onto plane and/or • because distortion, the number of patches will be too high eventually Levy, Petitjean, Ray, Maillot: Least Squares Conformal Maps for Automatic Texture Atlas Generation , SIGGRAPH, 2002 51

  52. Texture Atlas Generation • Split model into number of patches (atlas) • because higher genus models cannot be mapped onto plane and/or • because distortion, the number of patches will be too high eventually Levy, Petitjean, Ray, Maillot: Least Squares Conformal Maps for Automatic Texture Atlas Generation , SIGGRAPH, 2002 52

  53. Non-Planar Domains seamless, continuous parameterization of genus-0 surfaces 53

  54. Global Parameterization – Range Images 54

  55. Constrained Parameterizations Levy: Constraint Texture Mapping , SIGGRAPH 2001. 55

  56. Literature • Book, Chapter 5 • Hormann et al.: Mesh Parameterization, Theory and Practice, Siggraph 2007 Course Notes • Floater and Hormann: Surface Parameterization: a tutorial and survey, advances in multiresolution for geometric modeling, Springer 2005 • Hormann, Polthier, and Sheffer, Mesh Parameterization: Theory and Practice, SIGGRAPH Asia 2008 Course Notes 56

  57. Next Time Decimation 57

  58. http://cs599.hao-li.com Thanks! 58

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