Subdivision Surfaces

Transcription

1 Subdivision Surfaces 1

2 Geometric Modeling Sometimes need more than polygon meshes Smooth surfaces Traditional geometric modeling used NURBS Non uniform rational B-Spline Demo 2

3 Problems with NURBS A single NURBS patch is either a topological disk, a tube or a torus Must use many NURBS patches to model complex geometry When deforming a surface made of NURBS patches, cracks arise at the seams 3

4 Subdivision Subdivision defines a smooth curve or surface as the limit of a sequence of successive refinements 4

5 Subdivision Surfaces Generalization of spline curves / surfaces Arbitrary control meshes Successive refinement (subdivision) Converges to smooth limit surface Connection between splines and meshes 5

6 Subdivision Surfaces Generalization of spline curves / surfaces Arbitrary control meshes Successive refinement (subdivision) Converges to smooth limit surface Connection between splines and meshes 6

7 Example: Geri s Game (Pixar) Subdivision used for Geri s hands and head Clothing Tie and shoes 7

8 Example: Geri s Game (Pixar) Woody s hand (NURBS) Geri s hand (subdivision) 8

9 Example: Geri s Game (Pixar) Sharp and semi-sharp sharp features 9

10 Example: Games Supported in hardware in DirectX 11 10

11 Subdivision Curves Given a control polygon......find a smooth curve related to that polygon. 11

12 Subdivision Curve Types Approximating Interpolating Corner Cutting 12

13 Approximating 13

14 Approximating Splitting step: split each edge in two 14

15 Approximating Averaging step: relocate each (original) vertex according to some (simple) rule... 15

16 Approximating Start over... 16

17 Approximating...splitting... 17

18 Approximating...averaging... 18

19 Approximating...and so on... 19

20 Approximating If the rule is designed carefully the control polygons will converge to a smooth limit curve! 20

21 Equivalent to Insert single new point at mid-edge Filter entire set of points. Catmull-Clark rule: Filter with (1/8, 6/8, 1/8) 21

22 Corner Cutting Subdivision rule: Insert two new vertices at ¼ and ¾ of each edge Remove the old vertices Connect the new vertices 22

23 B-Spline Curves Piecewise polynomial of degree n B-spline curve control points parameter value basis functions 23

24 B-Spline Curves Distinguish between odd and even points Linear B-spline Odd coefficients (1/2, 1/2) Even coefficient (1) 24

25 B-Spline Curves Quadratic B-Spline (Chaikin) Odd coefficients (¼, ¾) demo Even coefficients (¾, ¼) Cubic B-Spline (Catmull-Clark) Odd coefficients (4/8, 4/8) Even coefficients (1/8, 6/8, 1/8) 25

26 Cubic B-Spline even odd 26

27 Cubic B-Spline odd even 27

28 Cubic B-Spline odd even 28

29 Cubic B-Spline odd even 29

30 Cubic B-Spline odd even 30

31 Cubic B-Spline odd even 31

32 Cubic B-Spline odd even 32

33 Cubic B-Spline odd even 33

34 Cubic B-Spline odd even 34

35 Cubic B-Spline odd even 35

36 Cubic B-Spline odd even 36

37 Cubic B-Spline odd even 37

38 Cubic B-Spline odd even 38

39 Cubic B-Spline odd even 39

40 B-Spline Curves Subdivision rules for control polygon Mask of size n yields C n-1 curve 40

41 Interpolating (4-point Scheme) Keep old vertices Generate new vertices by fitting cubic curve to old vertices C 1 continuous limit it curve 41

42 Interpolating 42

43 Interpolating 43

44 Interpolating 44

45 Interpolating 45

46 Interpolating demo 46

47 Subdivision Surfaces No regular structure as for curves Arbitrary number of edge-neighbors Different subdivision rules for each valence 47

48 Subdivision Rules How the connectivity changes How the geometry changes Old points New points

49 Subdivision Zoo Classification of subdivision schemes Primal Faces are split into sub-faces Dual Vertices are split into multiple l vertices Approximating Interpolating Control points are not interpolated Control points are interpolated 49

50 Subdivision Zoo Classification of subdivision schemes Primal (face split) Ti Triangular meshes Quad dmeshes Approximating Loop(C 2 ) Catmull Clark(C 2 ) Interpolating Mod. Butterfly (C 1 ) Kobbelt (C 1 ) Many more Dual (vertex split) Doo Sabin, Midedge(C 1 ) Biquartic (C 2 )

51 Subdivision Zoo Classification of subdivision schemes Triangles Primal Rectangles Dual Approximating Loop Catmull-Clark Interpolating Butterfly Kobbelt Doo-Sabin Midedge 51

52 Catmull-Clark Clark Subdivision Generalization of bi-cubic B-Splines Primal, approximation subdivision scheme Applied to polygonal meshes Generates G 2 continuous limit it surfaces: C 1 for the set of finite extraordinary points Vertices with valence 4 C 2 continuous everywhere e e e else 52

53 Catmull-Clark Clark Subdivision 53

54 Catmull-Clark Clark Subdivision 54

55 Classic Subdivision Operators Classification of subdivision schemes Triangles Primal Rectangles Dual Approximating Loop Catmull-Clark Interpolating Butterfly Kobbelt Doo-Sabin Midedge 55

56 Loop Subdivision Generalization of box splines Primal, approximating subdivision scheme Applied to triangle meshes Generates G 2 continuous limit it surfaces: C 1 for the set of finite extraordinary points Vertices with valence 6 C 2 continuous everywhere e e e else 56

57 Loop Subdivision 57

58 Loop Subdivision 58

59 Subdivision Zoo Classification of subdivision schemes Triangles Primal Rectangles Dual Approximating Loop Catmull-Clark Interpolating Butterfly Kobbelt Doo-Sabin Midedge 59

60 Doo-Sabin Subdivision Generalization of bi-quadratic B-Splines Dual, approximating subdivision scheme Applied to polygonal meshes Generates G 1 continuous limit it surfaces: C 0 for the set of finite extraordinary points Center of irregular polygons after 1 subdivision step C 1 continuous everywhere e e e else 60

61 Doo-Sabin Subdivision 61

62 Doo-Sabin Subdivision 62

63 Classic Subdivision Operators Classification of subdivision schemes Triangles Primal Rectangles Dual Approximating Loop Catmull-Clark Interpolating Butterfly Kobbelt Doo-Sabin Midedge 63

64 Butterfly Subdivision Primal, interpolating scheme Applied to triangle meshes Generates G 1 continuous limit surfaces: C o for the set of finite extraordinary points Vertices of valence = 3 or > 7 C 1 continuous everywhere else 64

65 Butterfly Subdivision 65

66 Butterfly Subdivision 66

67 Remark Different masks apply on the boundary Example: Loop 67

68 Comparison Doo-Sabin Catmull-Clark Clark Loop Butterfly 68

69 Comparison Subdividing idi a cube Loop result is assymetric, because cube was triangulated first Both Loop and Catmull-Clark are better then Butterfly (C 2 vs. C 1 ) Interpolation vs. smoothness 69

70 Comparison Subdividing a tetrahedron Same insights Severe shrinking for approximating schemes 70

71 Comparison Spot the difference? For smooth meshes with uniform triangle size, different schemes provide very similar results Beware of interpolating schemes for control polygons with sharp features 71

72 So Who Wins? Loop and Catmull-Clark Cl best when interpolation ti is not required Loop best for triangular meshes Catmull-Clark best for quad meshes Don t triangulate and then use Catmull-Clark 72

73 Analysis of Subdivision Invariant neighborhoods How many control-points affect a small neighborhood around a point? Subdivision scheme can be analyzed by looking at a local subdivision matrix 73

74 Local Subdivision Matrix Example: Cubic B-Splines Invariant neighborhood size: 5 74

75 Analysis of Subdivision Analysis via eigen-decomposition of matrix S Compute the eigenvalues and eigenvectors Let be real and X a complete set of eigenvectors 75

76 Analysis of Subdivision Invariance under affine transformations transform(subdivide(p)) = subdivide(transform(p)) 76

77 Analysis of Subdivision Invariance under affine transformations transform(subdivide(p)) = subdivide(transform(p)) Rules have to be affine combinations Even and odd weights each sum to 1 77

78 Analysis of Subdivision Invariance under reversion of point ordering Subdivision rules (matrix rows) have to be symmetric 78

79 Analysis of Subdivision Conclusion: 1 has to be eigenvector of S with eigenvalue λ0=11 79

80 Limit Behavior - Position Any vector is linear combination of eigenvectors: Apply subdivision matrix: rows of X -1 80

81 Limit Behavior - Position For convergence we need Limit vector: independent of j! 81

82 Limit Behavior - Tangent Set origin at a 0 : j Divide by λ1 Limit tangent given by: 82

83 Limit Behavior - Tangent Curves: All eigenvalues of S except λ 0 =1 should be less than λ 1 to ensure existence of a tangent, i.e. Surfaces: Tangents determined by λ 1 and λ 2 83

84 Example: Cubic Splines Subdivision matrix & rules Eigenvalues es 84

85 Example: Cubic Splines Eigenvectors Limit position and tangent 85

86 Properties of Subdivision Flexible modeling Handle surfaces of arbitrary topology Provably smooth limit surfaces Intuitive control point interaction Scalability Level-of-detail rendering Adaptive approximation Usability Compact representation Simple and efficient code 86

87 Beyond Subdivision Surfaces Non-linear subdivision [Schaefer et al. 2008] Idea: replace arithmetic mean with other function de Casteljau with de Casteljau with 87

88 Beyond Subdivision Surfaces T-Splines [Sederberg et al. 2003] Allows control points to be T-junctions Can use less control points Can model different topologies with single surface NURBS T-Splines 88 NURBS T-Splines

89 Beyond Subdivision Surfaces How do you subdivide a teapot? 89